The two games are very different, despite sharing the same underpinnings. I know plenty of people who played previous editions who don't like 4e, and I know plenty of people who played previous editions who loved 4e. Hopefully we can navigate these rocky, contentious waters without flames.
First off, 4e is fairly light on non-combat rules. This doesn't mean that 4e games are all about combat; it means that the rules assume that a lot of the roleplaying activities that were codified in 3e will be done via freeform roleplay. For example, there aren't any crafting rules for anything other than magic items. There also aren't any general professional skills, and there aren't any NPC classes. If you prefer to have rules for that sort of thing, 3e will be a better choice for you.
Second, 4e uses a power-based design methodology. Classes can be thought of as collections of powers; the differences between classes are defined by the different power choices they have. This makes for a very modular and flexible system. Some people find that it makes the classes overly homogeneous; some people like it.
Third, every 4e class uses powers. That's implied by my second point but it's worth mentioning specifically. A character begins with two at-will powers, that he can use whenever he wants; one encounter power, that can be used once per fight; and one daily power, that can be used once per day. Even martial characters, such as warriors, use this paradigm -- although their "powers" might be better thought of as something akin to a martial arts kata. This was intended to make combat more interesting for (say) fighters, in comparison to the earlier model where fighters just tended to hit things over and over again. If you didn't mind that model, this change may be unnecessary for your play style.
Fourth, 4e leans more heavily on the battlemap. My impression is that the large number of movement-oriented powers both make the battlemap more important and make combat more fluid, but that's definitely a subjective opinion on my part: consider it something to think about if you try 4e rather than a definite fact.
Fifth, 4e introduces the concepts of roles. Roles are a way of classifying classes by what they tend to do in combat. You've got leaders, who heal. There are more of them than just the cleric; for example, the bard is also a leader. You've got defenders, who control the battlefield by encouraging enemies to focus their attacks on them. The fighter is a defender; so is the paladin. You've got controllers, who are somewhat difficult to define, but you can think of them as the classes that affect the flow of a fight: they can hamper enemies, reshape terrain, and so on. The wizard is a classic controller. And, finally, you've got strikers, who purely focus on doing damage. The ranger and the sorcerer are strikers. Every class is primarily one role, but every class has the ability to take on aspects of another role, depending on what the player wants to do.
Sixth, multiclassing is more limited than in 3e. You can multiclass in a couple of ways, but you don't get the same ability to take six or seven classes/prestige classes during the course of your career. 4e classes are fairly flexible, but you don't get the same complete freedom you would with 3e multiclassing.
Seventh, the scope and feel of 4e can be somewhat more epic; or, to put it differently, more broad. The highest level is level 30, and that's very epic play, with abilities that allow characters to come back from the dead. Even at level 1, your characters are significantly more durable than third edition characters, and they'll be able to pull off some really wild things.
I think that hits most of the major differences. It's good to remember that it's still a heroic fantasy game in which characters fight monsters. It still uses a 20 sided die. Also, if you want to try it out, WotC has a free Quick Start kit available.
Alternatively, the new Essentials Red Box will be out in a few weeks; at $20 US, it might be a good way to take a peek at the game and decide if you like it. The Essentials core books will present a bunch of new class variants that change some of the things above: e.g., fighters won't have the same power structure I mentioned. So that might be a better entrance point.
This depends on a number of things. Do your players like to favor combat over other forms of role-playing? How long are your sessions? Do your players understand the rules well enough to where things are going to move by quickly (sounds like no, in this case)? Do you as a DM mind running a lot/a little/no combat in any given session? This sort of thing is highly customizable depending on your needs, and is thus unfortunately somewhat unanswerable.
Anyways, my personal recommendation would be to run a quick module or two before your main campaign. This will allow you to understand the sort of preparation a DM typically has for an encounter, as well as learning a couple of tricks here and there with DMing before you start your own campaign, which, in the long run, will be for the good of your story!
More importantly, it allows you to assess your player's desire for and prowess in combat. If it turns out they only make it through one encounter, then want to roleplay or ask questions or drink beer and relax for the rest of the session, you haven't wasted hours of preparation. But if they turn out to be some kind of death dealing blood-for-the-blood-god-skulls-for-the-skull-throne party, you are unlikely to run out of encounters! You don't even have to finish the module, really. Once you are confident in your ability to prepare adequately for your players, you're all good to go. Happy DMing!
Best Answer
Be Reactive
If you're just starting as a DM, don't try to adjust things in your first few encounters. Balancing encounters is something that gets a lot easier with experience, and if you get it wrong early you may just wipe the party out by overdoing it.
It also depends on how optimized the party is. If the party is full of experienced players playing very strong builds, you will need to throw a lot more at them than if it's a party of newbies or people just playing non-optimized characters.
Given all that, the best advice I can give you is for the first couple of encounters, do nothing. Treat the party of six like it's a party of four for encounter purposes. If you're using a pre-written adventure, just run what they have in it. If you're making your own encounters up, use what the Monster Manual and DMG suggest for a party of their level (as a party of four, not six).
If that turns out to be too easy for them, start ramping it up. You can do that in a few ways:
Personally I find that fights in my campaign go more smoothly if the enemy count is under eight. There was one encounter in the book that had like 30 Goblins, and it really wasn't that fun for anybody. So if I needed to make a fight harder that already has six enemies, I'm unlikely to add more and will instead make the existing ones tougher.
Even with only one enemy I'll sometimes still choose to advance it. If the party is fighting a big dragon, what other creatures can I add that both make sense and won't really skew the encounter? If the fight is three enemies on the other hand, adding a fourth can be a practical way to make it harder.