If you have Rick's Spain book, he has directions for getting from train and bus stations to the center, for all of your listed cities (as well as airport options for Madrid).
However, I agree that with four people, a taxi is the way to go. It may even be cheaper than other options (since a bus from the station is priced per person, while a taxi is per carload). I've never had problems with crooked cabbies in Spain (of course it's always possible). Just go to the taxi rank at the train or bus station, and show your destination to the driver. Do have it written down (last week I mispronounced my hotel by accenting the wrong syllable, and my driver genuinely had no idea where I wanted to go!). I see no need, and no advantage, to arrange for a taxi in advance.
For Madrid, from the airport to the center (where Hotel Fuencarral is) is a fixed rate of €30 including your luggage, and no tipping is required beyond this. I don't know if the Claridge Madrid is in the central zone where the fixed rate applies; your hotel will know.
For your other trips, it will be on the meter. Note that the taxis (except for that flat rate to or from Madrid airport ) will charge extra for each bag put in the trunk; this is not a scam. This and other legitimate extra charges (for instance, some cities have a surcharge for picking up or dropping off at the train station) will be posted in the taxi. Also, if you call for a taxi, the meter starts when the driver gets your call, so it costs a bit extra; to avoid this, you can go to a taxi rank. Ask at your hotel for the nearest one, since it may be close enough that you don't need to have one called, or it may be far enough that you do want one to come to the hotel's door.
Figure on 10-15 euros for most of your taxi rides from the station to your hotel.