If you're aiming above 60FPS (as in you have a monitor capable of 144Hz or above) and need faster memory support, I'd go Z370.
If you're locked at 60Hz, there's really no reason to go above 2666Mhz, so H370 or B360 would make more sense. Memory above 2666MHz will be forced to run at 2666 on anything other than Z370/ Z390 motherboards.
Unless you find a deal, a cheap Z370 board will cost at least $100. I found a combo deal on Newegg (i5-8400 and Z370P D3) and got the Z370P for $50 last year.
As for the best budget board, it all depends on what deals can be had at the time. Motherboards, performance wise, are all the same. If you're on a budget, choose the cheapest H or B series motherboard you can find with the features you want.