If I just bought new equipment for a business I just started, can I claim it on my taxes?

Normally, you can depreciate the capital investment.
You can take section 179 deduction.

Generally, the cost of a car, plus sales tax and improvements, is a capital expense. Because the benefits last longer than 1 year, you generally cannot deduct a capital expense. However, you can recover this cost through the section 179 deduction (the deduction allowed by section 179 of the Internal Revenue Code), special depreciation allowance, and depreciation deductions. Depreciation allows you to recover the cost over more than 1 year by deducting part of it each year. The section 179 deduction, special depreciation allowance, and the depreciation deduction are discussed in more detail in chapter 4 of Publication 463.

You will likely have to capitalize it and write off the depreciation (or use) of the equipment over several years.

By "use" I mean like some equipment is only good for so many hours, or miles, or whatever. In those cases you write off the use.

Otherwise, it may involves a MACRS depreciation over 5, 7, 10, or even 20 years. Read the tax codes, dude.

It depends on the equipment as to how/if it will have to be depreciated over the next few years.

If your company is not incorporated you should be able to