I'm sure there's a "How to gear up for your first LARP" somewhere.
Short answer - roughly $100-150+ to start off with.
Longer answer:
Here's a quick list of things:
Rulebook - $20 - this is the first thing for an Alliance game - even if you want to NPC, you should get one ASAP. They're on Amazon.com (including electric copy) or can usually be bought at the chapter itself.
Weapon - $15 - Using the weapon construction guidelines in the rulebook, and assuming you have such things as a hack saw already, you can make a decent weapon for under $15. Many people use 'advanced' weapon building techniques that use ultralight carbon fiber tube cores that would increase the cost by about $10 or you can get fancy and buy a latex weapon for anywhere from $60-$200 depending upon the shop. Note that if you NPC (Non-Player Character - you work as an 'actor' for the game, playing assigned roles and being the 'bad guys' that the players have to defeat), they'll have weapons for you to use.
Entrance Fee - $50 - this actually varies from chapter to chapter, length of event and such like. Usually though $50-75 is a price range. If you NPC though, usually the entrance fee is waived.
Food - depending on your appetite, just use the cost of what you think you'd need to spend on food for a weekend camping.
Costume:
This is the tricky part. It can be as elaborate or as simple as you like, and also depends upon your ability to sew. You can find many guides online on how to sew pants, shirts and even make period shoes. However, if you're looking just to buy, I'suggest finding a store that specializes in period items, shop eBay (using the term "LARP" in the search e.g. - LARP shoes) or look in the "Costuming and Props" area on these boards for people who make and sell such items. The following costs I'm taking from The Badger's Den - a LARP specific company that sells just about everything you would need for a game:
Shirt - $25
Pants - $25 (note a pair of black or dark sweats will work well for this)
Boots - This is tricky - finding a good and pair is something everyone is always asking for. Check the Costuming and Props section like I said - I'm sure a few are listed there. If you have a pair of non-descript boots or even moccasssins, they could work too. You won't be kicked out for wearing sneakers, just that it detracts a lot from the game.