My relative died and I inherited her iPhone. Her number was disconnected. Can I get a new phone number and a SIM card from a store? How can I start using this phone?
You may be limited in your choice of carrier, but as you surmise, the thing to do is visit a store. If the phone was locked to a specific carrier, you may need to get your plan from the same carrier, particularly if your preferred carrier is a CDMA carrier like Verizon. If you prefer to use a GSM carrier, the carrier can provide a SIM card and should also be able to unlock the phone from the previous carrier. (Provided, of course, the phone in question has a SIM card slot. I believe the original iPhone 4 did not if it was on Verizon or Sprint.)
The other caveat is that if it was running the iOS 7 software and your relative turned on the activation lock feature, you will not be able to use the phone until you gain control of her Apple ID. Going to visit an Apple Store may be able to help with this.