We have 15 days to emerge ourselves in Barcelona. Should we consider a fast train to Madrid? Would day trips to closer areas of the Barcelona region be better?Share with us any tips you have experienced, please.

Do you mean "immerse" yourselves?

Madrid isn't a day trip. Describe your interests and your lodging in Barcelona to get useful advice.

definitely-----immerse

We will probably stay in Sant Cougat at the #1 trip advisor selection of the AC. We enjoy visits to art galleries, historical sites and archeological wonders.

I recommend that you get some good guidebooks (or spend some time with a few in your local library), identify the things you most want to see in each location, note their opening/closing times, and mark them on a calendar. Then pencil in your transportation, add some time on either side (for getting to/from the train/bus station or whatever, checking in/out, packing/unpacking, getting oriented, etc.). Then see how things fit together.