It's hugely, hugely variable depending on the train operator but the links posted by MJR are a good way to check. Doing little 30 minute / 40 mile day trips near me is easy (I'm on Great Northern and they allow unlimited bikes outside of rush hour, I tend to get the train heading away from London at about ten am and then ride home from wherever, no need for a reservation).
Longer distances are usually slightly trickier (requiring a reservation in advance) but the resources above set the situation out comprehensively. There are one or two backward operators but the majority are bike friendly and largely speaking it's very pleasant to combine bike and train and extend your horizons a bit.
Cargo vans have gone the way of the dodo pretty much, but most modern trains have a few bike parking racks or spaces. On the commuter trains near me you just use the door area.