IPhone power button not working?

Should i take it to the verizon store or the apple store?

If you got it from Verizon, take it to Verizon. An Apple store attendant will just send you back to Verizon.

I'm presuming it's the iPhone 4 or 4S.

The two most common problems with iPhones (other than broken screens) is the Home Button and the Power/Lock Button. The power button is listed by repair technicians as THEE most difficult single repair in the iPhone. Everything in the phone needs to come out from the back, all the major internal components and finally the front screen before addressing the broken button.

The power button is actually a bubble membrane on a small flex circuit, where when you press the button it collapses the conductive bubble and makes contact. These usually fail if the phone is dropped on the button and the impact damages the plastic bubble on the circuit. The most reliable fix is replacing the flex circuit. Unfortunately it is tied into other parts of the phone and nested way up in a very difficult place to get to. It's partly glued in or clamped in and the technician needs a magnifying light work station to properly service the phone.

From personal experience, I can estimate the time about 1 hour of labor. It can be done faster (40 min?), but I've seen other technician's work, and it's cheap insurance to slow down a little to make sure everything is okay. The part itself is only about $10. Depending on where you go, labor is anywhere from $30 to $150 (Really!).

Things to consider: Is the iPhone still under warrantee? If yes, then take it to the Apple Store.
Check with Apple on line, or the store. If the iPhone was never registered sometimes the customer service will invoke a registration date such that the defective part can be fixed at little or no extra cost to you. If it's under warrantee and you have a unauthorized repair done, it will void the warrantee. If it's out of warrantee, then you have more options.

If the iPhone is out of warrantee, and unmolested, Apple has a buy-back plan where they will give you store credit for another iPhone. In the case of a bad power button it MIGHT be $120 (don't quote me, and the price is sliding depending on date purchased, grading (like scratches, dings and bad power button), memory. Sometimes you can qualify for a replacement (refurbished) iPhone.
If you choose to have it fixed, Apple will be the most trusted, but unfortunately, the most expensive.

Another option is to use a third party phone repair shop. AVOID the places in the big shopping malls. They do okay work, but their overhead is so much they have to charge twice what other shops will charge. I know a lady who went to the mall and they charged $160. That's about all an iPhone 4 is WORTH. Other places start about $50, most are around $75.

Another option is to SELL your iPhone. Do the Craigslist thing, discount the phone because of the broke power button. An iPhone 4 (healthy) on CL goes for about $170, so knock off $40 and sell it for $130. If it's an iPhone 4S, they sell for $250, so sell it for about $200.

Take the money and buy another (working) used iPhone.
I've only had my iPhone for 8 months, but I'm impressed with it. They are very rugged and hard to break.