If you are going grocery shopping, I have found it handy beforehand to get on to Google Translate, and write a list of all the products you want in Slovak before going to the store. In this way, not only will the labels be easier (if it's something not standard or obvious) and if you can't find something, you can show the word to an assistant to see if it's available.
Also, save plastic bags and take them with you. You can purchase bags but it is cheaper to take your own regularly. Also bear in mind you have to pack your own groceries here.
During my stay in Trnava (study, 3 months) I went shopping to :
Max: close to the train station you'll find MAX (by foot: walk in the park when you leave bus and train station (left hand side when station is behind you). After 100 m you'll see MAX on your right hand side. It is a small shopping mall (very nice and clean though). There are fashion shops, bars, a Chinese food shop, a cinema (cineMAX), sandwich bar, travel shop, electro shop,... At the ground floor, you'll find Billa (local supermarket).
Another Billa is located next to the city walls, near Student Street (don't remember Slovak street names however) and the Bernolak-statue.
Billa is opened 9.30 a.m.-8.00 p.m. 7/7 (except Easter,..)
At the other side of town (north-east) you'll find Tesco. It should be open 24/7 and you'll find everything here. There is more choice then in Billa. There is a shuttle bus leaving at the station for Tesco.
You can "buy" Trnava for 5 Slovak korunas :) in every single shop where are available post stamps ...