I would call myself a skilled player, I can rendevous and dock in my sleep, reach every planet and plan interplanetary gravity assists. However, I've never tried career mode, just because all the contracts seem tedious.
But, I did one anyway, and I noticed that besides science (which is useless without funds) going to another planet without a contract rewarding you for going there is pretty useless and pointless.
So do I have to follow the contracts in order to succeed? Because that seems quite boring and unexciting.
Best Answer
Technically, there's no need to go to any other planets in career mode if you don't want to — it's possible to max out the tech tree and upgrade all facilities just by sciencing the heck out of Kerbin and its moons and grinding contracts for those. (And, by in-game time, it's actually faster to do that than to launch even a single interplanetary mission!)
But ultimately KSP is a game of space exploration, and what's the point of playing it if you're not going to explore every place you can?
Also, even if exploration for its own sake isn't your main goal, sending probes to other planets can still have practical advantages. Besides the explicit contracts for it that you mentioned, there's also:
In particular, as long as you have a working probe with an antenna and at least one reusable science experiment in orbit around and/or landed on the body, you can complete these contracts just by accepting them, switching to the probe, running the experiment and transmitting the data. Even if you've already done the experiment before in the same biome and thus receive 0 science points from redoing it, it'll still complete the contract. Basically, it's free money.
(The thermometer works nicely for this, being the lightest and cheapest science module and requiring the least amount of electric charge to transmit. I like to stick a thermometer on all my commsats in career mode just for this reason. Also note that, even if you don't have enough batteries to transmit the data all at once, you can turn off the "require complete" option on your antenna and still complete the contract even if your batteries run out of charge during transmission.)
In modern KSP versions you can also get contracts requiring you to e.g. move a landed probe to another nearby location or to overfly a particular location with a craft in orbit. Depending on the capabilities of your probes, these can also be easy sources of funds, although it's always worth checking exactly where the contract wants you to survey first; sometimes they're not as easy as they look!
Anyway, if you're acutely strapped for funds in career mode, you may just want to take whatever contracts give you the biggest bucks for bang (and particularly the biggest advance payment!) and grind out a few of those. In my experience, exploration contracts actually tend to be quite good for that, although asteroid surveys and rescue missions aren't bad either (and the latter also give you free kerbals!). But it's perfectly possible to make an exploration mission pay for itself even without a contract, just with the funds you get from completing milestones.
Also, if you keep getting offered lots of contracts that you really don't like (such as tourists with awkward itineraries or base-building contracts with silly requirements or rescue missions from the surface of Eve), sometimes visiting a new body can be useful just to knock the contract system out of its rut and make it start giving you new kinds of contracts again. Of course, most of the new contracts will typically involve the body you just visited, so try to pick a body that you actually want to go back to, or at least try to send a long-term mission that can complete multiple contracts while it's there.