Although you can play an e-acoustic through a pre-amped electric amp, for e-acoustic with a integral pre-amp, IME a purposed acoustic specific amp does sound superior.
For your budget, typically you could buy a versatile Fender Acoustasonic 40. Despite my healthy scepticism of their prominent brand marketing, they do make acoustic amps worth considering IME. Reviews here and here.
At 40W, it'd have more than sufficient grunt to blow you away in your intended home 'stage'. It'd do fine for gigging typical small acoustic venues too, and comes with the bonus of a mic input with mixing controls for the instrument/vocal mix balance. I have Fender's smaller 15W sibling, which surprised me. I wasn't a Fender brand fanboy prior to purchase, and wouldn't say I was now, but I absolutely love how it sounds.
40W would be more than adequate for home use. I use the 15W Acoustasonic in my HT room cum home music studio which is a decent sized 4M x 4M (13½' x 13½') and carpeted, which deadens/absorbs the sound a little. Perfect for my e-acoustic APX600. 40W would rock an average house at full noise. I'd be confident enough to say 40W would even do fine gigging coffee shop type venues too at more modest volume typical single performer or background acoustic entertainment levels. I have three other electric amps to compare with BTW, a 100W (2x12") Fender Champion 100 modelling, 40W (2x6.5") Blackstar ID:Core 40 programable modelling, & ultra portable 6W Blackstar Fly 3+103 (2x3" in the 3+103) combo) as well as a Superlux portable PA system, a clone equivalent of Yamaha's since superseded Stagepas 300. 300W (2x8" woofers @150W) of noise mixing goodness.
If you've got an unamplified pure acoustic, you have a couple of options in regard to fitting a pickup, with or without pre-amp which alters your purchasing considerations. Alternatively, you could consider mic-ing it through a mixer, an all in one system like Yamahas Stagepas 400i or slightly cheaper Behringer Europort EPS500MP3 et al but that'd be well outside your intended budget. Fender make similar portable PA system series called Passport, but they are crazily priced comparitively.