The above posts are correct, and let me summarize (I sell paint for a living!):
First, my mantra: Clean, sand, prime, paint. Rest. Recoat.
Surface prep: Clean it up: Get some TSP (trisodium phosphate) and clean it all up. Wipe down your surface, free it of all dirt and dust and etc. Sand. You’ll need to etch up that surface. If there seems to be several layers of old paint on your current surface, you could try a stripping product (Jasco makes a good stripper for quickness - but it’s harsh and it stinks, then there’s a product called Ready-Strip which is greener but takes up to 24 hours to be ready to peel off). Another option is a deglosser (such as Jasco Paint Etch, or that liquid TSP, which is also a deglosser). The important thing for you to do is to etch up that old surface. Cabinets tend to be painted in satin/semigloss or higher, and that makes the surface “slippery” - new paint will simply peel away like rubber paper in a couple of weeks. Etching and priming gives your new paint something to adhere to.
Priming: Before you prime, it would be a good idea to find out if your old paint was oil based or acrylic. If you take a bit of denatured alcohol, put it on a clean (but old) cloth and rub it on the old paint - if some of the paint comes off, you’ve got an oil base. If nothing comes off, it’s water-base. What’s important here is that if you’re dealing with an oil base, you’ll want to seal it with an oil base primer. They stink, but thankfully, you can get some pretty good fast-drying oil based primers. (You can paint water base topcoat over oil base primer.) Seal Grip makes a great alkyd (oil) primer, as does Zinsser (Cover Stain is good, B-I-N primer is good for heavier damage surfaces, but I think it is shellac-base).
Topcoat: Some people swear by oil for wood and would use nothing else, but I myself recommend water-base paint. Oil tends to yellow over time, has no flexibility (wood naturally expands and contracts; the oil paint tends to chip off in splinters), it stinks (high VOCs!), and takes forever to dry (24-48 hours). Acrylics have come a really long way - many companies have a decent “acrylic enamel” - a water based paint that dries hard, like an oil. Often these are available in satin, semigloss, or gloss. I’d recommend satin as the lowest - semigloss is a good in-between - for durability and cleanability. Flat is chalky - doesn’t clean well and gets dirty quickly. Eggshell is a touch above flat - easier to clean, but better for living-area walls than for cabinets. (Actually, check with your local paint store on which sheen is which - some reverse what I call eggshell and satin, making satin the lower sheen and eggshell the higher. Be sure to double check!) The higher the sheen, the easier it is to clean. Satin is nice if you like low sheen and they don’t get heavy traffic/damage, semigloss is a great standard, and gloss is if you like to see your face in it!
Application: For oils, get a black china bristle brush.
For acrylics: Normally for trims, I’d recommend a stiffer brush. Acrylics are thick, and I’d point you in the direction of a poly-nylon blend. For a smoother finish, look for Chinex brushes. They are softer.
For shellacs: Get a white china bristle.
Purdy makes excellent brushes and won’t leave brush strokes in your paint. Angled brushes are great all-purpose brushes, for getting into those hard-to-reach areas and corners and junk; flat brushes are good for cutting in, for wide, flat surfaces and trim. 2-3 inch brushes are decent.
If you’d like to roll, don’t get too confused by the naps - it really depends mostly on your surface condition. Unless the old cabinets are really weird, they’re probably not very textured, and a decent 3/8" nap will do you fine.
Ask the salesperson about the type of paint you get - if it is an acrylic enamel, ask them about foam rollers for application. The paint manufacturer will be able to give you the best advice for that product.
Coats: depends on what colour paint you choose. For deeper, richer colours, you’ll need more coats of paint. Ask about having your primer tinted grey according to the LRV of the colour you chose (LRV=light reflectance value). It will help immensely. If, however, you have a white or relatively light, pastel-ish colour, white primer should be fine. I always recommend two coats (of topcoat), for uniformity and best coverage. Allow sufficient drying time between coats; read the instructions on your can of paint for how long is recommended for the product you choose. A gallon of paint generally covers about 350-400 sq. feet - one coat - and a quart covers about 60-75 sq. feet - one coat. Measure your cabinets before you buy so you know how much paint you will need.
Hints: “Condition” your paint tools before using them with acrylic paint. This is a fancy term for “get them wet”. 
Run your brush (or roller) under the tap for a moment, then give it a good shake off. You don’t want it soaking wet when you dip it into the paint, but a damp brush is good; you’ll avoid that “BLOOP” of the first brush stroke (the brush/roller hasn’t absorbed as much paint), making your strokes more uniform. This also makes it easier to clean your brushes/rollers when you are finished, ensuring your pricey equipment will last you much longer. These things are dirt cheap and help you clean up nicely. Oil paint will require a different cleanup method - you will need to look into getting mineral spirits to clean that stuff up (oils are so messy and inconvenient! blech! Use acrylics if you can!).
If you find the paint too thick to work with, ask the salesperson about Floetrol or Penetrol - you can thin the paint out to make it easier to work with.
Ask questions! Ask any friends you know, talk to any contracters you know. They often have invaluable advice. Good luck! It always sounds like a huge project, but doing it right the first time will save you so much time and money in the long run - and it will look beautiful when you are done.
Be sure to get everything you need for your project: brushes, blue tape for taping off edges (or decent equivalent, but be wary, don’t just grab the cheapest looking tape - the green stuff? Laquer tape - too STICKY! Masking tape? Ugh. Blue tape is often good, and ask the salesperson if they have a more economical tape for smaller projects - like 8 day tape or similar. It’s like blue tape, just don’t leave it up for longer than a week). Blue tape, be ready to spend a pretty penny per roll. Frog tape is new, it looks promising, but I haven’t used it yet.
Is that enough information? Think you can handle this? It’s really not as hard as it sounds. 
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