Hi!
Firstly, and not to put you off, you should know the RENFE website is the least user-friendly website there is, so don't despair. Having said that, it's your best bet to book these trips as you're going to be riding "their" trains... so it's always best to go to the horse's mouth. Furthermore, although train transport is semi-liberalised, RENFE is still "the" de-facto national carrier, so is as safe as it gets.
As Maria points out, not all credit cards are accepted, but this is not an issue just with this website, it's a problem you'll encounter with many other places here in Europe as not all US-issued cards are up to the new international security standards (just yet). It has to do with the existence of different standards within the US (local, regional and national banks...). If in doubt, ask your bank letting them know you're travelling to Europe. And if you have a paypal account do use it... you'll skip the issues I mentioned.
It's not your case, I know, as the itinerary you pointed is served by RENFE, but for other readers, you should know RENFE is not the only operator in Spain. There are regional operators as well. For example, here in Catalonia we have a public company called Ferrocarrils de la Generalitat de Catalunya (or FGC as we call it) which serves several lines alongside other lines served by RENFE. Note however each company uses different tracks (hence routes) and serve different cities. For example, the visit to Montserrat from Barcelona is served by FGC. FGC is a far better company than RENFE in all aspects: trains, website, punctuality... Unfortunately, though, high-speed trains (the ones mentioned by jjkoch4) are served only by RENFE -for now.
