Powered speakers weigh more but typically provide a better solution, with the integrated amplifiers being well-matched to the speakers and eliminating the need for separate power amps or for a powered mixer which, if it fails, leaves you dead in the water. If a single powered speaker fails, you ideally have at least one other to use, and probably your powered monitors, too.
You have to decide if you want to run your instruments through the PA or not. If you don't, you can use a smaller mixer and smaller speakers which don't have to reproduce bass frequencies. Without a sound engineer out in the audience, the mix is unlikely to be better or worse regardless of whether you mic the amps, go direct, or just rely on the backline to get the instruments' sound out there with each musician calibrating their volume to achieve a suitable mix. For smaller venues, I'd keep it simple and would not mic anything as long as everyone is restrained in setting their own volumes.
In addition to speakers and a mixer, you'll need stands for the mixer and FOH speakers, and cabling, so budget for those.
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