I would truly see if you can't find a decent used mixer board to begin. We're on our third mixer now, so I'm sure others are in the same boat. Don't be afraid if you're truly on a budget to consider a used mixer.
I would reach out to three companies here; Sweetwater (Cristian Good), Romeo Music, and Melhart Music. Melhart has a lot of things I wouldn't buy, but we bought a set of Yorkville Mains/Subs ten years ago that are still working well. Their carts for marching equipment also hold up quite well in my experience. If you ever watch any old Selena videos you'll see a lot of their brand of speakers in the background - worth checking out their brand name speakers maybe too. You might be able to get away with just one sub and one main if you're a smaller group and then get the second set next year.
For a mixer if you can look at the Yamaha TF1 would probably do well for you. I believe it will fit in a gig rig cart - which is the smallest size cart I would consider. Depending on your speakers you'll either need amps for passive speakers, or not for active speakers. They have their pluses on both sides. Passive mean you only have to run speakon cables out to the speakers generally, active you have to add power too. You can always tape the cables together though.
You could also look at an analog Yamaha mixer. That's what we started with originally and it works. The advantage with the TF1 would be saving presets and just going between scenes. You can also get a wireless router and control it with an iPad in rehearsals.
So that's mixer cart, mixer, speakers that I've hit. You'll need cables between all of those and amps if you go passive for speakers. The Yamaha amps are great, but availability for all yamaha things is not good. We ordered Sept 1 last year a new TF3 that didn't get to us until December. Those companies I mentioned may have some alternatives. Behringer used to be a really cheap brand, but there's a lot of corps/bands using their mixers. They're a bit cheaper and may be more available.
For mics with starting out the SM57 is probably the most bang for your buck. The AT 2035 is probably the best overall mic if you can afford them.
Be prepared to spend a chunk of change, but also be prepared not to buy this over night! We started with our SoundMachine speaker for a soloist one year. Then the next year we bought a gig rig and mixer that we ran into the soundmachine. The following year the two amps and the speakers were bought. Each year we've bought new things like a synth, mallet station, more mics, more cables, new mixers (we're on our third), etc. I'm most likely looking at doing new speakers next year I think. I've found it pretty much has to be a year-to-year investment and I know others are in the same boat.