Yes, you can buy UK train tickets online with the provision of collecting them at the station. The only problem is that ticket vending machines at stations typically only accept chip-and-PIN cards and thus with your foreign Visa / MasterCard, you may not be able to pick up your tickets unless it's a chip-and-PIN card. For long-distance journeys, thetrainline.com does has an international delivery option which costs an additional 7.50 GBP (although if you book early you can also get cheap 'advance' tickets).
If you do have a chip-and-PIN card, then sites like thetrainline provide you a unique one-time ID - this, along with your card details allows you to print your tickets out at the station itself.
For most short distance journeys, you can buy tickets at the station itself. This is not a problem when the station is staffed (during work hours), but can be when it's not as you will again find yourself unable to buy from ticket vending machines. You can buy your tickets on the train itself when the guard comes checking tickets, but the portable payment system they use usually doesn't support anything other than chip-and-PIN cards. Even with chip-and-PIN cards I have often found payments do not get cleared (PIN is accepted, but transaction is declined) so it's best not to rely on that. In that case you have to make sure you carry enough cash on you before you get on-board a train.
You may also be interested in GroupSave / group travel tickets that many UK train operators offer - you can find how much your savings will be by going to the 'train tickets' section on most train operators' websites. As mentioned in a previous answer, you also may find it cheaper to buy a multi-day / season pass.