Ice climbing beginner

I want to start ice climbing next season or maybe the one after, what equipment should I start buying first, are boots and crampons a good place to start?

Jack main wrote:are boots and crampons a good place to start?

Yup!

Go to a Festival (Michigan is close, I'm sure there are others in the NE.) and take a clinic.
Spend a couple days, try as many boots as you can.
Then buy.

Ygd.  Someone has to say it. Sorry. 

Buy boots that fit. Used boots are OK for now; you'll be lusting after something $$$ soon enough. Borrow everything else for the first few outings.

Join the AMC and go on a club organized trip.

Good fitting boots should be your priority. Ice tools and crampons can be borrowed or rented till you decide which ones you want to commit to buying. Make sure you have a harness, helmet and shell pants & jacket (goretex). 

Order of priority of gear to buy (IMO):
- Helmet
- Boots
- Belay puffy
- Crampons
- Hardshell or softshell pants (shotshell being my preference [and a popular preference] unless climbing super wet ice all the time)
- Tools
- Hardshell or soft-shell jacket (see comment about about pants)

By this point you should have enough experience to decide if you need/want fancy gloves, or belay mitts, or other misc pieces of clothes. You may also be thinking about leading by this point, and that's a whole other thread for gear you might be interested in.   

I wouldn't spend a fortune on boots right away unless you know your feet well enough that you know when you have a good fit for a pair of boots, and feel pretty confident you know what direction you want to go in your ice climbing (alpine, single pitch cragging, mixed climbing, etc). Many people climb in boots for years that either don't fit them or suit their style of climbing because they can't stomach spending the money on a second set (albeit better suited) pair of boots.