This is a can of worms as everyone has their favorite printer, but here is a bit for you:
Print and cut always sounds great up front. but the reality is that printers that can do both don't really cut well. A few years back we added Graphtec 9000-160 plotter and it increased our ability to print - by allowing the printer to continue to print instead of stopping, loading the laminated piece and cutting. So, if in your budget, add a good wide format plotter (like the Graphtec or a Summa).
Keeping the budget down, Mutoh and Roland Eco-Solvent printers are a great start. We use a Roland XR-640 and it has been a workhorse for about 8 years now. We do not have odor issues with this solvent printer, and the inks hold up well to natural UV light without fading. For long term
signs and for wraps we do laminate with a good cast wrap laminate.
For better quality, Epson offers some great printers that will hit a wider range of colors.
For top quality and speed, the Cannon M series are great. But these are not a good choice for a small shop as they are a bit pricey unless you are printing in volume.
We use a Cut Work Table CWT1640 to cold laminate, mask, and even to apply vinyl to
sign substrate (like acrylic faces, MDOs, ACM composite).
Some printers do put off an odor and you will want to see them in person before buying.
I am not a fan of Latex printers and there have been plenty of discussions about them on here you can look up. I am sure someone that loves theirs will post something, so will let them mention the positives and negatives.
In any printer choice, it is best to have your own job files to try printing. They differ in color spectrums as to matching colors. Usually greys, beiges, reds, and oranges are the hardest to match.
If your budget is low, buy a Mutoh or Roland print/cut machine.
If your budget allows, buy a Roland or Mutoh print/cut machine and a Graphtec FC series plotter. If your budget is a bit larger, get the Summa plotter instead.
An great setup for a small shop would be the Epson printer and either the Graphtec FC series or Summa plotter.
If you have a big budget get the Cannon M series and one of the two plotters.