Use caution when using MDF....while dimensionally stable, it will swell up quickly when exposed to moisture....fine as long as you can keep the inside of the garage dry.
Having done all of it, I no longer make drawers. It's cheaper to call up one of a couple of drawer shops and have them make me what I want. When you consider the cost of the material and labor involved...it's just cheaper.....at least for the small to medium ones. The labor for a small drawer is about the same as a large one...so I only make the large ones. For a garge, Dovetail is not necessary. You can buy from Rockler a router bit that makes joining the corners a lot easier. If you are making your own, make sure the bottoms are at least 1/4". If you go to the right place, you can get plywood specifically made for that....the layers are a lot thinner and overall it's stronger and more stable than standard plywood....for some reason the name is eluding me right now....not enough coffee yet...
For the rest....I would suggest Mellimine. Easy to use, does not need painting. If you make the doors out of it, make sure you seal the edges with the iron on edging...that will keep out the moisture. You can also get a few stock colors, white, gray, brown and black. Also, make sure you get the good stuff...the one with the plastic covering...not the paper....
For hanging the doors, you want to use something like a Blum hinge....which will require a special tool to drill the pocket in the door. Rockler sells them for about $20. Once you have it...you will love it.
Don't get hung up on how to hang the cabinets (pun intended). On the inside at the top, have a 2"H x xxW strip that runs across the back. This strip is attached to the top and sides...just screw through it into the studs on the wall using #10 screws. You will have a similar strip on the bottom of the cabinet. The bottom one carries a majority of the weight and the top one keeps the top against the wall. 4 #10 screws and you can expect it to hold 200-300 lbs.
On the bottom you want to make what is called a face frame cabinet. Basically it's a frame made up of 1.5" wide strips of wood. The below pic is an example of one I did for our bathroom. I would 'highly' recommend another purchase from Rockler....Pocket Hole cutter... Trust me on this one. It's the easist and strongest way to join the parts of the face frame together. You can build up your face frame screwing it together but no glue. Once you are happy with the final result, take it apart then use glue along with the screws. For the face frame wood I would suggest Poplar....NOT pine. Much harder and stronger...and only about 25% more in cost than pine...
For the drawer slides....spend the money and get full extension slids. If it were a kitchen...I would use the hidden softclose type...but for the garage, use some heavy duty types....you don't care if anyone sees them....
A couple of #'s for you.
24" is the standard depth of a counter top.
36" is the standard height of a counter top and works well for a garage. If you are taller, go 38-40".
16" is the standard distance between the counter top and the bottom of cabinets. If you make the upper cabinets deeper than 12" then you need to increase this distance as well...say 1" for each 1".
12" is the standard depth...but as noted above, is sometimes not deep enough. Personally, I would not go deeper than 12"...at least for the bottom part of the upper cabinet. Any deeper and you will feel like the cabinet is 'in your face' when you are standing in front of it...plus it reduces your view to the back if you are standing up.
Since you are planning on going all the way to the ceiling, may I suggest making the upper 16" deeper...say 16"...the bottom of the door will be above your head....or at least should be.
Hope this helps/confuses you.