First: You want a true sine wave inverter if at all possible, Several reasons for this, but mostly it is very clean power.
The smallest I've seen TSW inverters is 300 watts. this is enough for your television and computer, May run the house radio/home theater.. or not (not sure it would run mine).
For you, at this time, that is what Id suggest.
Kick it up to around 1500 watts you can microwave lunch or use a hot plate, But you need a lot of battery to do that (About 200 amp hours per kilowatt of inverter).
Mine is 2000 watt, does all the above at the same time.