I did A LOT of research before attempting to paint my furniture. I grew up in a house where that was taboo and you did not put color on wood and veneer was absolutely not paintable. I guess that's why I kept waiting for someone to walk in and yell at me while I was doing this whole project! Anyhow, this is what I learned through trial and error and trying a few different things:
First, Zinsser cover stain is your best friend for this project! Note, this is oil based and doesn't just wash right off your hands and brush when you are done. In fact, make it easy on yourself and just buy a cheap pan to toss afterwards and you use cheap brushes for this project too so plan on throwing them away as well. And buy this baby in the gallon size because you will use it a lot once you see how easy this project is!

The other crucial items you need are FOAM brushes... I repeat FOAM!!! Zinsser picks up all and every fiber off a regular rolling brush. I also feel you need a small foam roller, a large foam roller and the giant bag of craft foam brushes for those hard to reach places and cracks. By the way, these are really inexpensive at Walmart. As much as I detest Walmart, it's worth the trip for the pan to throw away to the brushes.

Okay, next item you need is sand paper and wood putty. Here is the truth behind sanding... you don't even need to unless the piece is damaged. I know this because I tested this theory out. The bed I painted had been chewed by a dog and I sanded the crap out of the headboard. But, the foot board I left untouched except where the dog ate it. All I did was fill the holes and sand it after it dried for a smooth look.
CHEWED UP SECTION:

WOOD PUTTY BEFORE SANDING IT SMOOTH:

Now that you know the basics, one it comes to your final top coat of paint here's another tip. I went to Benjamin Moore and bought the paint all the websites recommended. It was $20 for a pint!!! I flirted with the paint man and that was even an employee discount! (Shhh.. don't tell J!) So I painted one dresser with this special expensive paint and one with the Behr Ultra Premium paint which runs $30 a gallon.
You do need two coats of paint so the gallon is what you will end up buying. BUT... as long as you use TWO coats of Zinsser and 2 coats of the top paint... it doesn't matter! I had to use two coats of both types of paint and they look identical. This goes for the furniture I sanded like mad and the ones I was too tired to sand and just said "Eff it"! AS LONG AS YOU YOU A FOAM ROLLER, YOU DON'T NEED EXPENSIVE PAINT!!!! I ONLY USE FOAM ROLLERS!!!!
My last tip before you get started with this entire project is cleaning the furniture before you paint. Many people said to go with TSP. It's waaaay more expensive than my homemade version. I just put equal parts of white vinegar and water in a spray bottle and wiped it down with that. It took so much crap out of the wood! I did wipe it down before and after sanding too. You will be grossed out by the dirt it pulls out!! For the TSP... this is non toxic and works amazingly!
So here's the run down one more time:
1) clean
2) sand if you must
3) clean if you sand
4) 2 coats primer
5) 2 coats of paint
AND PLEASE PLEASE PLEASE... wait at least 8 hours between coats so you don't get a tacky piece of furniture. This parts sucks the most!!!
Here are my before and after pictures: (eventually I want to replace the hardware but I'm keeping it as is for now)
BED BEFORE:

BED AFTER:

DRESSER BEFORE:

DRESSER AFTER:

OTHER DRESSER AND TEASER OF E's ROOM: