I'd recommend a Yamaha DBR10 powered speaker. You can add a speaker stand later when you gig. Also, a small mixer like a Yamaha MG10XU, which has a fantastic delay effect in it. There are two channels with built in 1 knob compression and 4 total XLR/combi connectors. Also, this small mix desk works as a nice little interface for recording. Yamaha makes an optional (mic) stand mount for it. You'll need a mic cable (male XLR to female XLR) to hook the speaker to the mixer. I use this system outdoors to facilitate a weekly event at a campground which has up to 100 guests. The speaker is very light, yet has enough punch to cover a large area if needed. A 10" is about a perfect speaker to still be able to handle bass and drums, and still sound good for vocals and acoustic guitar while remaining compact and lightweight. A big plus to this route is that a) the components are pro level gear that will last a lifetime, b) it's super expandable. Down the road, add a second speaker 1 for monitor, one for FOH [to the audience]. c) you can get very loud [if needed] sound without it breaking up, unlike some of the self contained PA towers/systems out there costing much more. d) you'll have money left over to buy an effects pedal or two
The 3 band eq should do an adequate job to shape your guitar tone.
When jamming at bandmates' homes, the DBR10 by itself makes a fantastic "monitor" for our keyboard player.
Going this route, there are many many choices for mixers and powered speakers. Read the reviews, see if I'm wrong with these as my picks. Cross check prices too.