Vegas has some of the finest dining in the world, and there are so many restaurants you could visit a new one every day for years and never run out of places to go. However, sometimes you just want to go to a grocery store.
The closest supermarkets to the Strip and Downtown are on Maryland Pkwy around the UNLV area Ė which is only 5 minutes or less east of the Strip. The first is Vons on Tropicana and Maryland, next is Vons on Twain and Flamingo, and between the two and on the other side of the street is an Albertsons.
There is one pharmacy on the Strip close to Monte Carlo, Bellagio, and City Center, and there is one right in the middle of Downtown on Fremont. If you are looking for local prices, take Flamingo east to Maryland Pkwy on the northwest corner in the Target shopping area.