I agree with the comments above that this question is a little too broad, but "college student's budget" and "limited space" give at least a starting point and I can offer a few pointers based on that.
Not only for budget reasons you're looking at primarily a hand-tool approach.
In addition to the cost of power tools working with hand tools is quieter (so people who share a floor or walls with your room won't be disturbed by loud motors late at night) and will generate less dust. Hand-tool processes tend to generate flakes and shavings and little dust (which is also coarser), much easier to clean up than large volumes of fine dust as generated by power sanding for example. Fine dusts are also more of a health hazard if you're working in or adjacent to where you sleep.
You need a good solid surface to work on. There are a couple of simple approaches adopted for this for those without a dedicated working space, one being modification of an existing piece of furniture (e.g. a stout kitchen table) with a lift-on/lift-off woodworking surface. The second is by building a smaller knock-down woodworking bench which can either fold flat to be stored leaning against a wall, or break down into pieces which can then be slid under the bed, behind the sofa etc.
Two examples of the first approach are given in the following images:
Source: Woodwork magazine.
Source: LostArt Press.
And a good example of the second is this, the Apartment Workbench:
Source: Close Grain blog. Full build instructions and parts list at this link.
Some discussion on the requirements of the casual woodworker can also be found in a classic book on woodworking which is available free on Project Gutenberg: Woodworking for Beginners, by Charles G. Wheeler. This book is a wealth of information for the hand-tools-only approach to woodworking so well worth downloading to your hard drive to refer to as you like.
Now to tools. There are a great many tools you can get but what is required is probably mostly driven by the nature of what you're making. The above book gives a good idea of the hand tools required for a large amount of work to be successfully carried out, but you don't need all of these immediately.
One common piece of advice given today is to buy tools as you need them for specific projects, eventually building up a collection that will allow you to do a wide range of projects. Buying tools secondhand, e.g. from flea markets, car-boot sales, estate sales, thrift stores and sites such as Craigslist is a good way to get older and vintage/antique tools that can be extremely good value (often better made than modern equivalents) and can perform excellently with a little cleanup and fettling.
My minimum list for smaller projects would be something like this:
- at least one saw
- some cheap chisels (e.g. from Harbor Freight)
- low-angle block plane
- some sharpening supplies for the chisels and the plane's blade (commonly referred to as an iron)
- a strop (can be made very simply and inexpensively)
- a drill of some kind and a range of bits to drill holes in common sizes
- a pencil
- steel rule (both for measuring and for use as a straightedge)
- a carpenter's square
- one or more card scrapers (minimise the use of sandpaper)
- some clamps (the more the better, literally buy as many as you can afford)
- woodworking adhesive
- abrasive papers (wet & dry types are not intended for wood but can be washed clean when they become clogged with sanding dust and dried for re-use)
Sharpening supplies are a must. Most edged woodworking tools are not supplied sharp and must be sharpened before first use, and regularly maintained afterwards to keep them properly sharp. A sharp tool works better and is actually safer because it requires less force in use. Periodic stropping is the best way to maintain sharpness quickly and easily.
I would list these as desirable but not immediately essential:
- a marking knife (a craft knife or boxcutter can work adequately here)
- a smoothing plane (Stanley type no. 3, 4 or 4 1/2)
- a bevel gauge
- a marking gauge (this can be made by the woodworker, many plans available for free online)
- a flat-bottomed spokeshave
Rather than this answer running even longer, ask a follow-up question about jigs that the woodworker can make for improving the efficiency of handwork operations and I'll provide a pictorial list. This will include a bench hook (sawing aid) and a shooting board (planing aid).
