(my first post on this site - be gentle)
Minimum needs: a place to grow, a container, a bag of soil, seeds, and lights (fertilizers are mentioned below too)
Place: can be free, but 1) make sure you have plenty or room for plant size and lighting fixture(s), and 2) note that at a certain point in the growth, it will smell. Be sure you have adequate ventilation and that wherever you grow, you have a way of safely dealing
with the odor (such as a fan out the window). Best bet for the future, when you've got a little extra money - a grow tent + a flexible aluminum vent/duct + an inline duct fan, with the hose going directly out a window.
Container - can be free, ideal to have little cups for starting seeds, then consider 3-5 gallon pot size such as a large 5-gallon bucket, a large cat-litter container, something like that. Make sure it's got holes for drainage and a pan/tray underneath to catch overflow
Soil - you will read plenty of discussions about this brand or that brand, but for a minimum start - just get a bag of 'potting soil' at your local Home Depot or WalMart or garden store. You'll see chatter about vermiculite and perlite, but go with a decent potting mix and you'll likely be OK
Seeds - 'bag seeds' are ok, but keep in mind that it's a bit of a crap shoot AND you have a 50-50 shot of having male plants result - and male plants have to be thrown away. If you can buy feminized seeds from a seed bank - it's a great investment. But, if we're talking about saving money, you may try the bag seeds. You will need to grow a couple of plants all they way into the flowering stage and it can be frustrating to have to discard your 'babies'. My first grow I tried 4 plants from [rather old] bag seed and had to throw away 3 of the 4 plants. My one ultimately worked out well, but you get the idea - for cheap, use bag seed. For best results and your own satisfaction - buy from a reputable seed bank.
Oh - someone is bound to jump on this 'Hey, what about nutrients'? Well, what about them? Here's the deal with soil grows - with decent potting mix, you don't HAVE to add any nutrients, although something as simple as Miracle Grow or some similar 'houseplant' fertilizer will do wonders. Before I get scolded for this - the ideal situation is that you have a nitrogen-rich fertilizer (even like a Miracle-grow type of product) for the first half of the grow, and a 'bloom' fertilizer for the second part of the grow (which would be a relatively low nitrogen fertilizer). Your plants will still grow and will still work out for you without any fertilizer (assuming you've got good soil), but let's plan some cheap nutrients, maybe $10. I assure you, if you plan for a future grow you will almost certainly build your nutrient collection - but you don't have to overspend!
So far the total cost for this BARE MINIMUM setup is - Place: 0 Container: 0 Soil: $10 (or less) Seeds: 0 Nutrients: $10- Now we get to the last item - lights
Lights: What you probably want to do is go out and get a 400-watt hood, ballast, and 2-bulb package, plus you'll need a light timer and some mechanism to suspend the light fixture. BUT... that set-up just for the light, timer, cords, etc. will eat up most of your total budget In my opinion, if you have the space and adequate ventilation - I'd just go with the 400-watt HPS/MH package. You can also go with a smaller, and cheaper size, like a 250-watt and that would be adequate for a small grow. Another option that is attractive is a compact fluorescent set-up. My suggestion is to NOT go with a bunch of individual household-size bulbs but rather to go get a T5 fluorescent tube fixture (if you elect to go with CFL). One problem is that you then may need two sets of bulbs, one for the 1st [vegetative] phase and some other color spectrum bulbs for the 2nd half (flowering) stage. You can easily spend well over $100 for that fixture and bulbs (I think mine was closer to $150, but you might be able to go with the 2-foot version for $100 or less?). You probably would supplement that with 2-4 clamp-on lights and CFL bulbs, as once the plants get kinda tall, the CFL tube fixture doesn't really cover the lower portion of the plant all that well.
Anyway, most of your budget will go toward lights, and if you have inadequate lighting then everything is for naught anyway. For the budget, maybe T5 fluorescent 4-bulb by 2-foot long OR 250-watt MH/HPS is OK for you. People will debate this forever; another possibility is clamp-lights and UBER-big CFL's. I'm not talking about 23-watt, or even '150-watt equivalent', I'm saying those bulbs that are perhaps 85-watt (labeled as '300-watt equivalent'). Those bulbs run around $15 -$20 each. You could get 4 of those with clamp-lights but I say better yet - get a T5 fixture + a couple of those 85-watt GFL bulbs OR get the 250 or 400 watt hood/ballast combination. Long story, but if I had the chance to do it all over again, I'd just get the 400-watt unit and be done with it (example here - but I claim no knowledge of the particular brand, it's just an example:
http://www.amazon.com/Apollo-Horticulture-GLK400GW19-Digital-Dimmable/dp/B004YXDHX6)
Shopping List (did I forget anything?):
Seeds
containers
soil
fertilizer(s) - ideal includes a grow fertilizer, a bloom fertilizer, some molasses (read about it), some others only if problems arise (calcium, cal-mag, epsom salts are some possibilities)
pH meter (optional, but someday you'll need it)
bug control - Insecticidal soap, possibly neem oil, possibly yellow sticky cards
Light timer(s)
extension cords
rope, chain, or [better yet] rope ratchet for suspending lights
Tent (optional)
Fan(s) - ventilation
Duct + inline duct fan (if you can, or if you go with a tent option)
Spray bottle (for misting plants)
Carbon filter system (maybe in the future)
Lights
Oh - and most important of all (lol) - trimming shears, Mason jars, and Bovedo 62% Humidipaks!