Tell Them Your Goals
If you haven't already, I would start by telling them essentially what you just said here. That there is no "one true plot". Tell them that introducing an evil person / problem does not make it the overriding campaign unless they want it to be. Tell them that you are willing to follow along with their character's background goals.
If they want to play that style of game, there is a good chance that will fix a lot (but not all, habits are hard to break) of it right there.
But its worth remembering that not everyone wants a sandbox. I prefer to play in games with good plots with well laid out characters. While there are some exceptions, GMs frequently are willing to invest more time into plotting and developping good NPCs if they are assured the PCs will focus on them. For that reason, I am perfectly happy to with what you might call "light rails". I want some character autonomy, but I am perfectly happy to follow "the one true plot" if it gets a better story and better NPCs.
By telling them your goals directly, they will either agree to go along, or directly ask you to put some rails back and then you can choose as a group how much of the game should be on rails.
Drop many plot hooks, make them choose
In the game itself, one way to force them to avoid rails is to drop so many potential plot hooks they have to choose which ones to focus on. You can even show that the world is a living breathing place by showing them the resolutions to the plot hooks they didn't take.
For instance, they sit in a bar and overhear of some great evil. They may be tempted to go after it immediately, and there are no rails, so they can! But then someone comes up and asks them for help urgently. And I don't mean Skyrim style urgently where everything is urgent but will wait for you without ever changing, give them an actual deadline measured in something like game days.
Now, they have a choice. Go after the big bad or help this person? In a game on rails, the answer is first help then go after the big bad. Obviously, the person asking for help is just there so they can level up. But you can break that by making it clear the big bad isn't standing still. If they go after him now, he won't have completed his McGuffin of Death and they can stop it before its an issue. But if they wait, they will have to deal with him and the McGuffin. Now, they have a real dilemma and rails won't help them.
To take it even further, highlight something unresolved or some goal on the character sheet. Point out that the world isn't standing still and any time they spend either helping this person or going after the big bad won't help them deal with that personal (to the character) matter. It too may get harder if they put it off.
Time is a great limiting factor in real life, it can be in games too.
Now, they have choices to make and no rails to help, they really have to choose. That is the first and biggest step.
After they get used to dealing with that, especially if you keep highlighting personal goals from the character sheet (and insist they add some if there aren't some obvious ones) they will eventually start adding more of their own, like becoming a dragon slayer. Then they will start actively trying to achieve those goals without further prodding.
The word that comes to my mind is 'ALLIES'.
The ally is an NPC that wants (or needs) the group. They can help in three ways
A) Offer themselves to the group as a wandering helper. A person who wants to travel to the city you are going to will welcome the extra security.
B) Offer them equipment to help the part. It could be a character's parent (mum!) passing on a family heirloom magic sword, etc.
C) Offer them help with contacts throughout the realm. If an old wizard has asked the party to gather ingredients from another town, then there is no reason that the wizard doesn't have friends in the other town. Someone roughs you up when you go to pick up the the ingredients, and suddenly the local wizard guild comes to the rescue.
My rule of thumb for lightening a session up is to have the NPC do two things: be funny with an outgoing personality that makes them very likeable, and allow them to crack a joke at the best (or worst!) of times.
The other thing this NPC needs to do is to somehow show a display of power; if your brother joins the group and he is a peace loving monk then have him beat up the local bully, afterwards he turns to you and says 'they did teach us to fight as well as flower arrange you know ... '
Best Answer
You could present the game in a more reactive fashion.
If the players do not appear to be making any kind of indications of the sort of actions that they would like to take then you could simply present low-importance information back to them and again prompt them for action.
For instance: The players would like a means of entering a city and have their reasons not to do so in conventional ways (through the front gate - through the guards).
You could start be describing the city as being... "...on a slope, with most battlements appearing to be well-maintained, the grey slate mostly devoid of growths and only the faint shadows of the sun high in the sky betraying the existence of minor weather erosion. The breeze blows warmly across the landscape and activity at the gate seems calm."
In the lack of an indication of action you could prompt the group with the soft suggestion - "Would you like to explore further?"
(Side Note - if the response is something as low-brow as "we wait a little to see what happens" then an appropriate response could be "You wait a few minutes and watch as a small flock of pigeons rise and fall in the distance over the rafters... Nothing of further note catches your attention. What would you like to do now?" This presents the low-importance information mentioned earlier - unless there is good reason to add information (a time-line like an approaching storm or a nearby threat))
If yes then "OK. Tell me who in the group is going to be exploring and in what manner. Are you looking for anything?"
And in response to the lackluster "Looking for a way in" - "OK - You need to tell me specifically what kind of way in you are looking for and how you are going to do it"
This will engage the players and certain avenues may present options to you. Perhaps scouting the walls in a careless or unstealthy manner might result in a challenge from atop one of the watch towers. Or perhaps they will happen upon a small group of shady characters with which interaction might occur.
You may also be presented with options that you had not considered.
A search of the walls for a means to scale it might reveal a particular location between battlements where external plumbing has been affixed externally to the wall, leading downwards and terminating a short way above a lonely sewage wagon. This would present the possibility of an easier climb DC should they choose this way to enter - although there is no guarantee that it will not be detected or that it will be safe. The small stream inlet further north around the city (assuming that the character searches further) might prove a better bet - especially if you can swim.
Although the players needn't be told that unless they test the waters first - that is part of the fun of being a GM - you don't need to worry about common sense on behalf of the players - you can provide warnings of difficulties if you wish.
In this way you are presenting the kinds of choices that the players elicit from you. If they don't appear remotely interested in going through the gate then there is no need to present the hint (if specifically searched for) that the guard might accept a payoff - or that disguises might work due to the apparent lazy disposition of the guards in general.
Of course you will occasionally want to prod the group if they end up in a situation with no clear way forward. If boredom is setting in and the group is stuck looking for posters of jobs at the guard house then perhaps their attention might be drawn to a short old man who watches them intently from outside the door. That may play out in various ways - whether to present a task or to present a new foe or to direct the players to a location that they would be more likely to find work.
Alternatively it is possible to draw upon the characters' background history - a little clue that occasionally pops up here and there. Whether it materializes as expected is not important - it helps to provide direction and direction is a precursor to momentum.
Presenting secondary events
When you spend time designing secondary plots for the party to stumble upon it is understandable that you would 'wish' for the party to stumble upon them even if they might not otherwise do so.
While you could certainly place all these encounters (conveniently) in the path of your players, there are several factors to consider:
Would it be economical for this basement-dwelling dabbler necromancer to 'move' to a run-down shack in an unnaturally silent part of the forest (think 'Evil Dead')? Or perhaps a next town if the concept would work better in an urban environment.
One last possibility is introducing the possibility of consequences for events that are not encountered and tackled. Perhaps failing to notice the dabbler necromancer would result in an increased incidence of undead attacks in the surrounding country-side as his poorly contained energies seep away and awaken a worse evil than he or she could ever have imagined. This could also apply to the potion shop ruckus - a few ill-mixed ingested potions could release a crazily mutated humanoid upon a rampage.
Suspiciously peaceful aftermath - If your players encounter nothing but resistance from varying seemingly unrelated forces before completing the main event, and emerge to sunshine and rainbows within a suddenly peaceful city then one or two players might begin to suspect that one or more secondary plots were contrived - placed before them - possibly denting the suspension of belief that goes with good role playing games (its not so likely if they actively pursued reasonably challenging and scarce clues). Even if you don't happen to have any good secondary plots left for this city you could always fall back on the consequences of the breakdown of the structure of order in part or all of the city (The guards are distracted. Looters ahoy!).
Player conditioning towards proactive approach - If the players do not express any desire to explore the rest of the city then they do not have to. Once again you can recycle materials for other locations in terms of shop descriptions and the like (just be sure to keep track of what NPCs and places you put in which locations - jot notes on the fly). Furthermore you could occasionally tease the players by having a couple of travelling NPC casually comment between themselves that "The 20% sale was a novel idea", or that the "Artifact auction was popular" or that "Villagers have started disappearing in the night", etc.
Eventually the players may get the idea that it is worthwhile to get a small idea of a city as you never know what you might find.
Hint - If you are not into detailing maps then you could do worse than to set all the locations (or clusters of locations) on a probability chart. If they roll successfully for a particular location then you could describe that section in shallow detail similarly to when you described the city at first sight - let them explore and direct the flow. Rerolls can attempt to avoid areas already explored - alternatively a good streetwise test (even characters can get lost in some places that are big enough).
Conservation of energy as a GM
While some of these thoughts may hopefully prove useful to yourself in conducting your games, remember that they are a set of suggestions that need not be implemented in full. You will find it desirable to determine how deeply you wish to describe your world and an excess amount of it may be pleasing but it may also entail a greater degree of creative involvement on your part. This is why I mentioned certain time-saving elements like use of pre-determined probability charts - these are tools intended to take a little of the burden off your mind - but should not serve as an obstacle to your doing what you wish to do (you can fudge a roll or allow direct movement if it takes your fancy) - after all, the tools serve you, not vice-versa.