Traveling with my wife and 2 year old daughter to Spain next month and plan on relying on a lot of public transportation and walking during the trip. Will we be able to navigate without too much trouble around Spain between the metro, train, and bus systems?

We plan on arriving in Madrid and traveling to Segovia, Salamanca, Toledo, Seville, Cordoba, Granada, Murcia, Valencia, and Barcelona via buses and trains mostly.

Will we be able to do this trip with a two year old over a two week period? Will she be welcomed at are museums?

Any ideas on a good travel path/hotels? Thanks in advance

If you stay in the centre of Madrid, there is a lot of things in walking distance. Some metros are more stroller friendly than others, I havent been on the buses, but im sure it wouldnt be too much trouble. There are a lot less baby changing facilities than here in the UK, but you can still get by.
The Spanish are very child friendly and they will be welcome in most places, and you will see whole families out late at night as well.

That seems to be a lot of travelling to go to all those cities in a 2 week period, I would maybe look at what you really want to see and cut out half of them.

I travel when my kids were 6 mos old, and no bigger problems than home. yes eskimo joe is right, depends on the metro station, the newer ones are equipped, the older dont.

but you will get help, children are king/queen in Spain. And yes two weeks traveling with the children in all those cities might knock you out.
hope it helps

In Barcelona you will find that the buses have allocated space for strollers and wheelchairs and on the Metro too. I don't recall allocated space for strollers on the trains, but there is enough space unless your stroller is unusually grand.

People here are usually courteous enough to take into account that you are travelling with a stroller an will move if they are occupying the allocated space. This is a generalization and you may have to ask though. So at least in this city you shouldn't have any problems.

Hope that helps.