My team builds out of my garage and as such our equipment is very basic. I would recommend you have: (warning: this list may be excessivly long)
At least 1 bandsaw (12" or 14" min.) Have spare blades.
At least 1 drill press
2 or 3 good battery powered hand drills
1 air-hammer hand drill
1 table saw
1 or more jigsaws
1 Sawzall
2+ Dremel tools
4 or 5 complete sets of wrenches (metric and standard)
1-2 Sets of screw drivers (Phillips and Flat head)
5-6 ratching socket wrenches and several complete sets of sockets
1 or 2 Ratcheting Screw drivers
Several sets of screw driver bits
1 complete set of TiN drill bits
3+ sets of High speed steel drill bits
1-2 Counter Sinking Tools
1-2 Deburring knives
Tap Magic (lots of tap magic)
2-5 vices
Lots of clamps (QuikGrip and C-clamps)
1-2 Right angle T's
1 set of Files
Several Taps and Dies
Sand Paper (maybe even electric sander)
1-2 Rubber mallets
Several Hammers
Millions of Zipties
Duct Tape (preferably military-grade duct tape)
2-3 Robogrip Pliers
2-4 Channel Lock Pliers
2-3 Vice Grips
1 set of Needle-Nose Pliers
1-2 Tin Snips
Several pairs of Wire Cutters
1-2 wire stripping tools (the nice ones that hold the wire and strip it)
1 good crimping tool
1-2 soldering iron
1-2 Good Chain Breakers (#25 and #35)
1 Hand Rivet Gun
1-2 Exacto knives
Several Hacksaws
1 Miter Saw and Miter Box
Several cans/bottles of Dust-Off, WD 40, Goo-Gone, Loc-Tite
A scale (we use a Fish Scale, the ones for weighing fish)
You could try a small table-top mill or lathe (we haven't really had a need for one though)
I am sure I am forgetting something but this is good enough to get started with, plus you can get most of these parts pretty cheap if you buy in sets.
One of the places I have had the best luck finding cheap sets of tools is Costco and Wal Mart (even though I prefer Craftsman but those aren't as cheap). I bought a complete set (100-150 pieces) of TiN drill bits and Circle cutter things for about $70. The only problem is they do not always have this stuff, I think the best time to check is around Christmas. I also would recommend Skil as a good brand of hand power tools. I bought a set of from them having a Hand Drill, Circular Saw, Sawzall, Light, and various attachments for about $200. The saw had a laser guide, drill had lights to illuminate drilling area, and the other tools had other cool features.
I cannot think of anything else right now. The great thing about having a lot of hand tools is that you can get all of the team to participate in building which is not always possible. We do this on my team and we have 25-35 students.
For nuts and bolts I would reccomend you buy sampler kits from McMaster because that way you will get a decent sized assortment of many sizes of bolts, screws, and so on; plus these generally come in a nice box that is already sorted which is great for organization. Have lots of spare nuts, bolts, wire, screws, connectors, and other parts of various sizes and types. This way you can always have parts on hand.
Another good investment is buying lots of those fancy fishing tackle boxes. These boxes are great for sorting parts because they have tons of slots but also are portable. These are great for competitions because you can carry everything you will every need pretty easily.
You can buy shelving or build it. Building shelving can be a good way to teach new kids how to use drills and saws, as well as keep them busy at practices.
I do not know what kind of computer would be best but I would recommend looking into some nice workstation or gaming computers (for CAD of course), I prefer laptops but desktops generally are better. I do not know if your sponsor does this, but many companies have contracts with companies like DELL or HP and recieve special discounts on purchasing computers and software. Your computer does not need to be all powerful to work well enough for FRC (it does help), I use a 5 or 6 year DELL Inspiron 6000 and have inventor running "fine" on it.
Hope these help.
P.S. sorry if I rambled a bit.