Usually the countertop is screwed down from inside the cabinets. You
should see little triangular brackets in the corners with screws going
up into the counter. Take down the measurements and go price a
replacement laminate countertop. They can even cut the sink hole for
you cheap.

As for the backsplash, since you want a temporary spruce up, forget the
tile backsplash. They sell this stuff that is made out of hardboard
that looks like tile. It has some kind of enamel finish with genuine,
imitation, simulated grout lines. Comes in 4x8 sheets and you'll find
it with the paneling. It cleans up easy and is very water resistant,
It will also cover any damaged drywall if the counter backsplash is
glued to the wall.

In our first house we did a kitchen clean-up that consisted of:
Painting the cabinets - White Muralo - easy to clean, great finish
Replacing the dark grey countertop with a nuetral green laminate
replacement
Replacing the chipped & stained porcelain sink with a SS sink
New faucet
Put the *tile* paneling up as a backsplash and behind and around the
stove
Less than $500 for everything

Wasn't going to make it to the cover of Better H&G but looked a hell of
a lot better than the battleship grey phenolic we ripped out. Three
years later when we sold the house everything still looked great.

If you do replace the countertop take the time to prime the bottom and
around the sinkhole where the exposed particle board is. Also use a
ton of silicone under the sink lip and between any countertop joints
and let it squeeze out everywhere then clean up with vinegar before it
sets up. Hardest part of the whole job was getting the double-bowl
cast iron sink out by myself. A 2x4 across the top and heavy rope
through the drain holes held it up while I removed all the clips while
laying underneath it.