Pick a mixer with enough mic channels, and preferably at least a couple of aux outs. Behringers are just fine, I have only positive experience with them actually. I'd also recommend powered speakers, and you can use a budget brand here as well. I have a Behringer Eurolive B212A that I use as a monitor, and it sounds at least half decent. It is a bit bass heavy, but there's luckily an eq on the back. You can't compare the sound these produce with the higher end stuff, but will get you started. You can buy higher end active speakers later and start using the Behringers as monitors instead.
Speaking of monitors, I'd recommend having at least one of them. You can manage without them but then you must place the main speakers behind you on the sides of the stage so you can hear yourself through them. That is however not really possible in all venues.
You could manage without a subwoofer, but that's only if your amp is powerful enough (at least a few hundred watts) and you don't feel a need to mic the kick drum. Otherwise I'd recommend getting one or two of them as well.