Spain with a toddler in tow

Spanish people are REALLY attached to their kids, much like it was here in North America 40 years ago. They take their kids everywhere with them,....in fact, don't be surprised to see children wandering around the popular restaurant districts with their parents at 1 or 2 o'clock in the morning on weekends. And the extended family still plays a great role in activities in Spain.
For children your daughter's age, there are a couple of great places in Madrid. Two are in the southwest part of the city, in the large Campo de Casa park. Their zoo is one of the finest in Europe, and includes an aquarium, a "dolphinarium" and an aviary. There's also a small model farm, where kids can get close to the smaller farm animals. They hold shows daily involving the water animals and birds. It's closed on Sundays.
In the same area is the Parque de Atracciones, a huge amusement park. While it contains lots of wild rides, there's also a zone for small children to enjoy. It's usually open only on weekends after mid-September.
Both are easily accessible by Metro line 10, Batan stop.
Of course, you can't pass up Retiro Park, even if just to let her burn off some energy. If the weather remains nice into your trip, you'll probably still see a fair number of street performers near the lake.
You could also plan a day trip to Segovia, easily reachable by high-speed train from Madrid's Chamartin station. There are a couple of wonderful massive old merry-go-rounds that will enchant her!

One other suggestion...since you're going to be a week in each city, considering renting an apartment. It's nice to have a place to call "home," especially with a toddler. We've used homelidays.com with good success in Madrid and elsewhere.

As Norm says, children here are involved in the family to an extremely high degree. My daughter (18 mo.) has been accompanying me to bars since she was three months old! (While this might sound outlandish in the U.S., it is extremely typical here for the neighbors and friends down at one's local to get to know the kids. Now that some bars have become smoke-free it is easy to find a place where a child will be welcome.) You will be hard pressed to find a place which will not be kid friendly to one degree or another. FNAC and El Corte Ingles (Madrid, Barcelona, Seville, etc.) offer free, clean changing rooms. FNAC has a kids reading nook. Ikea has a number of kid spaces (especially valuable if it rains or if the nipper needs to recharge batteries. As far as Barcelona specific stuff: My kid loves the lights of the Magic Fountain, the beach, the playgrounds at the Ciutadella Park, the playground in the large community space between C/ Carders and C/ Sant Pere Més Baix on the corner of C/ Metges. (About 5 min walk from the Palau de la Música, in a neighborhood well worth exploring.) Barcelona is filled with tiny playgrounds (often just a slide and a swingset). Take advantage of your kid's playtime as an opportunity to break the ice, and talk a bit with the "natives" in the parks. Feel free to IM me if you need other, more specific advice.

Thanks to you both for your thoughtful replies! Spain does sound like a great place for our first "over the pond" vacation with our daughter!

I think you are smarter than we are. We did our first 3 trips with toddlers to Germany/Austria/Switzerland -- in Spain you will be able to 'embrace Siesta / nap time'. When we traveled with little kids we felt like we were giving up a 'prime part of the day' to park somewhere and let ours get in a nap.

...as we will be traveling with our kids to Spain in 2011... I'd like to hear more about kids in bars. Do I understand right that there is no problem if we take our kids with us for drinks / tapas? Even late into the night? Are there any places that are inappropriate for kids we should be aware of?

Jim, I suppose that a brothel wouldn't be very kid friendly... Seriously, kids are seen here as part of society, and as such are tolerated and welcomed just about everywhere. Especially in central and southern Spain, it is not infrequent to see kids playing around in plazas where their parents are having tapas/drinks at 1-2am. Please understand that I'm referring to social/tapas bars, not discotheques, as those are all 18+ (except for the earlier, kids session).

Now, if I could just get my daughter to understand that she lives in a siesta culture...