The inverter can be a cheapy from harbor freight or the like. I actually think costco has a decent inverter. There is no need to go for the pure sine wave with the devices you are looking at.

To start with, look at how many watts you are looking to use. Most devices have a peak power rating or some other form of power rating. For each device decide how many hours you think you will need to power it and then total it all up. This is your expected energy usage. Then I would oversize by about 20% to compensate for losses, etc.

Example:

So say you have a 12w cf bulb,

12 w x 8 hrs = 96 watts total.

So your inverter would need to be about a 20watt, and then you say how many amps you need total and per hour, so you say have a requirement of 12 watts at 12vdc = 1 amp/hour or 8 amps total.

So basically you then use those numbers to find a battery that meets your requirements.

For your needs you could probably get away with a 500 watt inverter and a simple deep cell battery for a car.