Some time since I did it (nearly 10 years ago) but I think the basics are still the same.
Most people take the Shatabdi Express, which leaves from the main New Delhi Station at about 0600. That gets you into Agra Cantonment (the best stop for the sights) at about 0800. The train back, from memory, is at about 1700 (at least the one I took was - tho' I seem to remember I got the ticket changed as I'd had enough of Agra by mid-afternoon; other than the Taj and the Fort, it's a bit of a dump - an industrial city with a population of just over 1m).
Several travel agents can make bookings for you. I was in Delhi on a work-related visit so our office travel implant fixed it up: you could try Amex or Travel House.
A few words of warning:
- When you book your ticket, they'll ask you your age. I've never quite been sure why, tho' I suspect it's got something to do with concessionary rates. And when you get to board the train, you'll find that your name and age are posted up on the side of the appropriate carriage for all to see...
- The air-conditioned coaches on the Shatabdi are bitterly cold, so take something to wrap up in!
- It's worth considering booking a guide to meet you and take you round at the other end. If you don't, you can be sure there'll be no shortage of willing volunteers waiting for you at the other end who will take you round the sights, then drag you round the carpet shops, marble shops, gem stores...
- You're likely to have all kinds of weird 'security' conditions imposed upon you at the Taj and the Red Fort, like having to hand in cigarettes and lighters, pocket calculators, etc. At least Kay did when we went in 2002. (However, she did get them back.)
- Again, there's no shortage of unofficial 'guides' at the main tourist spots. They're likely to try to charge you hundreds of rupees for quite trivial (and probably unwanted) 'services', like showing you good places to stand for photos. My advice is to ignore them (as best you can). If you want a guide, there are plenty of official ones who know what they're talking about.
Having said all that, it's worth a visit, just for the experience - but it's striking how many people who've been say they prefer the Fort to the Taj (which, for all its craftsmanship and beauty, is after all a tomb with a story rather than a dwelling with a history.)
HTH - have fun!