If its real ceramic tile and in good shape, no loose or cracked pieces then you can go over it as long as height is not an issue. If its anything else do yourself a favor and remove it all down to the plywood subfloor. Tell us what is left along with the size and spacing of the floor joists and their unsupported span. Your answers will determine whether yor subfloor is strong enough as is to support a tile floor without them cracking.

On the counter put down the 1/4" cement board after you paint on a liquid waterproof membrane around the sink opening and anyplace else that might see water, embed the cement board in a troweled layer of thinset and anchor the board to the plywood with galvanized roofing nails.

The house is on a slab, no plywood flooring. im just gonna take my time an take up any excess on the floor to make it look right.Any kind of sealer would work to put on the counters before putting the backboard down would be ok, right(thompson water sealer)? What about putting a sealer on the concrete floor? Thanks

STOP!!! NO SEALERS on the concrete. It will inhibit the bond of the thinset and just use the waterproof membrane around the sink opening on the countertop. No seaqlers there either. After the tile and grout are installed you can use a grout sealer to protect against stains.