Fitz, thought it might be useful to recommend some of the products I would start with when getting into mixing on a budget.
1. Gin (an absolute must in my book but I really like gin). There are plenty of good gins around. I really like Plymouth or Bluecoat but they cost around $25. There are plenty of good gins under $20. For a good budget gin, many go with Gordon's which is close to $10 but I haven't had it in a long time so I cannot verify its drinkability.
http://www.nytimes.com/2007/05/02/dining/02wine.html
2. Vermouth (dry and sweet). I might be inclined to get a bottle of Dolin blanc instead of a dry vermouth and just cheat the recipes a little because it is so damn good. However, it cost around $15 and there are certainly cheaper alternatives. Noilly Prat is popular and goes for around $7 per bottle. Again, it has been a while since I have had this so I will let someone comment if they have a better suggestion. I would stay away from Martini and Rossi though. Also, keep these in the fridge and they will last longer.
3. Rye. I love Sazerac 6 yr but it will run you $25-30 and there may be a cheaper alternative that is just about as good. Wild Turkey Rye is also supposed to be very good. Stay away from Old Overholt. It is cheap ~$13 but tastes like "panther piss." You could also substitute bourbon but then Al and others will give you hell for making a Manhattan with bourbon.
4. Bitters (Angostura and Peychauds)
6. Stuff like lime, lemons, tonic water, as needed.
7. Campari, not too expensive ($15) and will allow you to make a Negroni. I would get this after you have the other stuff though as it cannot be used in as many drinks as the other stuff.
For the future:
Herbsaint or Absinthe if you want to make the delicious Sazerac. Absinthe can be pretty pricey though, it does go a long way though.
Brandy
Anything else I am forgetting?