The six skills of the game are cooking, mechanical, creative, body, logic, and charisma. If you're planning on getting to the highest level in each career, then you'll need to get all of these skills to their maximum level (10); otherwise, you may want to just specialize in one or two skills while letting the rest lay fallow. Each career's highest promotion requires a 10 in at least one skill, however, so you can plan on having your Sim do some serious studying in the latter portions of the game. There are multiple items that your Sim can use to advance his or her rank in a skill; the more expensive the item, the faster your Sim will advance. Your Sims will refuse to work on skill improvements if they're in a bad mood, however, so satisfy their motives before you put their noses to the grindstone.
The Counter-Culture career track is the only one with a substantial investment in the cooking track; it requires you to completely max it out. None of the other careers require a skill level higher than four. Low-level advancement in this skill is accomplished through studying cooking via a bookcase (all of which advance you at the same rate, by the way), while Mimi's Place has a couple of unlockable stoves which will give you a higher rate of advancement, and will also let you create food which can be served at parties or eaten by your own character. At its highest ranks, though, Cooking is the hardest skill to learn quickly, so you may want to start dedicating a little time to it early on.
Although you can increase your mechanical skill by studying at a bookcase, your best bet is to buy a Tesla Coil 3.0 and use it for rapid acceleration in your learning speed. If you can't afford to spend almost $9,000, though, you can settle for second best and pick up the Kraftking Inventor's Workshop at $3,330. The Workshop also lets you create salable gnomes; select these in the Buy menu and sell them off to earn back the cost of the Workshop itself. The price you'll get for the gnomes will increase as your mech skill improves. Your mech skill will also improve your chances of repairing any broken items around your house.
Your Sim's score in the creative skill can be somewhat important, and not just for satisfying promotion requirements. If you choose to build up your creative muscles with an easel or an Artist's Block, for instance, you can sell off the art that you create for profit; the higher your skill, the more money you make.
If you're not interested in moolah, though, a Tumultech "Sim-Phony" Synthesizer is the fastest way to get a high Creative score, and also gives you a big boost to Fun while you play it. Until your Creative score gets to around five, though, you can expect to offend anyone who listens to you and lose relationship points. You can unlock this item at the Goth Manor, as well as find one built into that house; if you plan on practicing on it, move it into an out-of-the-way location before using it to make sure that no one can hear you.
Every career except for the Mad Scientists will require at least four points in the body skill. Your options for body-sculpting range from the lowly exercise machine all the way up to the $8,000 dollar Abdominator, which is unlockable at the Octagon. The low-level exercise options are pure skill-increasing machines; the more expensive items will get you a small Fun boost while you work out. Your Body skill will rarely affect gameplay outside of your careers, but if you initiate an Attack Social option on another Sim, the one with the higher body score will emerge as the victor.
Your logic skill has no bearing on any gameplay matters except your career advancement, so don't bother studying up unless you're looking for another promotion. The most expensive item in the game (Miss Gyrotic, unlockable at the Octagon) is a logic study item (among its other functions), and is great for quick skill boosts if you can swing the 10,000 dollar price tag. The other options are the telescope (unlockable at Shiny Things Labs), which gets you a bit of Fun while you stargaze, and the more expensive Gene Genie Incubator (unlockable at Goth Manor), which breeds mutant plants which you can sell.
Charisma is the most taxing skill to master, as there are precious few skill-building items to buy, and those that are available don't allow you to double up on your skills and motive replenishment, with one exception. The most basic skill-building items are mirrors, which your Sim can stand in front of and practice a speech with. Beyond those cheapie items, however, is the Nostalgico Microphone ($1,100, unlockable at Tinsel Acres), which lets you sing your way to a high charisma, and have a bit of Fun while doing it. Its price belies its relative ineffectiveness at building your skill, though, so you can expect to spend a lot of time belting out the oldies before you gain to 10 charisma. As a bonus, though, other Sims will listen to you sing, and if you're decent at it, any listeners will see their Fun and relationship scores with your Sim boosted.