Whatever you’re buying (not renting) should be the best quality and capability you can afford. Think channel inputs, mix outputs, routings, effects needed, etc. There’s nothing worse than committing to a mixer, then later finding yourself needing things it can’t deliver. Hence the “ buy once, cry once” philosophy… get what you need now and later, as best you can figure, and you won’t find yourself shopping again in a year.
Basic/bargain gear is fine if you have basic needs and expectations. With your stated needs, I wouldn’t buy anything with less than 24 inputs, 32 would be even better insurance against getting “channel-choked.” For that, and additional aux mixes, I’d say the X32 Rack is the best option among the suggestions made here. I use the Presonus 32R, and have used the Behringer X32 and 32Rack. All solid, compact, roadworthy choices. The ability to save scenes and mixes is a great time saver playing at different venues repeatedly
Powered mains are the way to go for small bands traveling a lot. Go and personally listen to what’s available and pick what sounds best to you. I’d say to size them to your regularly expected crowd and plan to rent some extras if you score a bigger gig.
Powered monitors also work fine, though IEMs all around can help you hear better while helping bring down stage volume, something important in more and more venues these days. Too-loud bands rarely get the callback, so be able to match venue expectation.
Splitter snakes are the norm if you have to feed your rig and a house system. Sometimes you can just give them a summed mix and they’re ok (assuming your band mix sounds good). But on the times when they want all channels at their board, a splitter gets you there, and your monitor mixes (a big deal for IEMs) come from your mixer, as usual.
Research is your friend, as is enlisting the help of a local player or sound guy who knows his business to help you select purchases available in your area. I think RCF is made in Italy, and is a good speaker brand. Perhaps look their way.
It’s good to get input from the forums, but there’s no replacement for putting in the footsteps to go see what others are using in your vicinity, and what it sounds like, and if that band is using a similar amount of ins and outs, to help you settle on equipment that will really deliver the bang for the buck you’re looking for here.