How do I update my kitchen? It’s 6x6 tile and needs some attention

For the back splash you could go with the inexpensive stick on tiles.

For the counter top, you could paint it.

Remove the tiles. Sand, prime, and paint walls with 2 coats good quality kitchen paint. You can also opt to install laminate, butcher block hardwood, quartz, stainless steel, soapstone, granite, etc. on counter. You can place larger tiles on walls, or subway tiles, or paint reverse side of glass, and mount pieces as your backsplash. If you want to save cash and time, there are artisan tiles in 6"x 6" size that you could place in random pattern, amongst existing tiles, if you carefully remove select wall tiles, and can match grout.

Removing any tiles is not an easy, fast, or clean job. If you have sheetrock/drywall behind the backsplash tiles now, the damage caused to the sheetrock to the walls when the tiles come off might very well be too much to save the existing sheetrock. If you are wanting the backsplash area painted, the sheetrock will most likely first have to be replaced with new.