How much are you looking to spend?
For errands/commuting a huge thing to consider is if your area is safe to leave it locked outside or if theft is a major concern. If there's significant chance of theft you don't want to invest TOO much because that's just more you stand to possibly lose.
Generally, get something with brazed on bolts for mounting racks. I'm running this set up:
Topeak Super Tourist rack with matching trunk bag. The MTX system is very nice for the bag/matching rack. I can fit a fair amount of groceries on there, shopping for one person and not needing a giant load of shit its very doable with the panniers. Great for picking up a six pack or bringing a growler home from the brewpub. If you are, like, buying like 12 packs of soda and cases of water or big bags of dog food or shopping for a family of three plus....well good luck doing that on a bicycle really. There's also always backpacks for extra load, and you can do front mounted racks/bags for more capacity on the bike. I personally don't like a lot of weight on my back while riding but that varies from person to person, and it depends what sort of ride home you are dealing with as well.
Its on an old (2007?) entry level Trek road bike that had braze-on's in the rear for racks. I have a beefier CX wheel/hub with a 25mm gatorskin (this means nothing to you right?) in back for more durability/strength under cargo load but I could probably have gotten away with stock too.
Ao- is going to tell you to get Salsa or Surly, so I'll just jump the gun on that. Salsa's Vaya or Fargo are built to carry loads, and Surly makes a number of touring bikes that are purpose fitted for bags as well. If you are looking to spend more and have an actual nice bike (and are confident it will be safe locked to a rack) these are really your go-to companies. The Surly Troll is a cool bike that's a bit more on the mountain bikey side.
Otherwise, any old road or mountain bike that can easily accept a rack or racks is the way to go.