We are traveling with our two kids (ages 3 and 1) in mid-May to Barcelona for about 5 days. We won't have a rental car so I would ideally like to stay near a metro stop, close to a grocery store, and also close to a lot of Barcelona's attractions. I'll have a double stroller and would like to be able to just get up in the morning and easily walk with the kids. I know I won't get to see everything there is to see because we are time restrained (naps, baths, bedtimes, and temper tantrums), but we're going to do our best to see Barcelona's high lights in the short amount of time we have. Any suggestions on which neighborhoods to stay in and/or out of? Thanks for your help!

Get an apartment. Go to www.apartur.com, which lists vacation rental agencies in Barcelona. We used BarcelonaStuff.com and had an apartment just off La Rambla in the Barri Gotic within 2-3 blocks of Placa Catalunya (the heart of the city) and across the street from a small grocery store. The apartment building, which was quiet, was at the end of a small alley accessible only through a locked gate, which kept the foot traffic to a minimum.

Another option is: get a street map, look at the box formed by the Placa Catalunya, Tetuan, Verdaguer and Diagonal metro stations and stay at an apartment in that box. You'll be pretty close to a ton of sites and you'll be close to all the main Metro lines (Nos. 1-5). Do not pick an apartment company outside of Apartur - that umbrella group does try to maintain some standards.

And if Barcelona is your last stop in the Euro-zone, pick an apartment that does NOT require a cash deposit because on your last day you'll be stuck with a pile of cash that you'll have to exchange at bad rates.

Second the apartment idea. We stayed at the Cadaques apartment run by Destination BCN located in the Eixample district. Fantastic location and apartment. 24 hr mini market located a few doors down. 10 to 15 mins walk to some Gaudi buildings, La Boqueria, Plaça Catalunya, Barric Gothic district, etc.

If you will be using a stroller for your kids, take a look at this interactive metro map to check if the station near the apartment you like is "accessible" (activate the elevator icon)
http://www.tmb.cat/ca/transport-accessible

Buses are usually a good alternative during off-peak hours as most sights are in a quite compact area of Barcelona. So going down and down and up and up to the metro for just one stop can be more tiresome than to just wait 5 minutes for the next bus.

Was definitely planning on getting an apartment so I'll check out those sites for a good fit. Thanks! Also, good tip on buses vs metro. Didn't think about a stroller going up/down so many times. That's also a reason I should probably want to stay central to lots of activity. Less time trying to figure all that out.

A good fried of mine studied in Barcelona in college and said to stay in Gracia near the green line of the metro. What do you all think of this? Is it too far out? She was in college at the time so is it "collegey" or more for families?

paseo de gracia area is NOT far out. Very nice area. When you find some apartments you like , those familiar here will be able to steer you towards the better neighborhoods for what you are looking for.

perhaps the OP is referring to Gracia area... which can be a bit far out.