Location was a major factor on which grocery store I use when in LV. There is a Von's, Albertsons, and a Target on the east side of the strip. They are all between the strip and Maryland Parkway.
You will want to sign up for their "card" to get the sale prices. Was not sure if you were used to that coming from the UK.
I'll probably be staying at palms place, so figuring it as my base.
I did get caught out by this card thing once, buying vitamins from somewhere once the register rang up to about $650 since I didn't have a card, the woman behind me swiped hers instead and brought it down to about $250!