Barcelona’s metro system is fairly simple to negotiate. Frequent trains and clearly indicated arrival times make travelling by underground a fast and simpler option. Escalators are usually available but for those with prams, lifts (elevators) are few and far between. Main interchange stations will usually have lifts although you may have to walk a way to find them. Look for the sign that has a box with people in it when you get off the train.
More lifts are being introduced. The Purple Line L2 has lifts at every station. People do tend to offer help to those struggling up steps with strollers but it can be quite tricky getting to the platforms let alone getting through some of the narrower gates. Put your ticket through and indicate to the ticket office person that you have a pram as there is usually a special wider gate for easier access. If not you may need to go through the exit gates the wrong way if your pram is wider than average. See a list of Barcelona metro stations with lifts / elevators.
There are various travel card options that will enable you to get the best deal for your family. Children under 4 travel free. After that they pay the same ticket price as adults. The T12 ticket for 4-12 year olds is a travel pass available to Zone 1, resident children. Find out more about the Targeta T12 in Barcelona