You can get train times from here (along with advance purchase tickets):
http://www.nationalrail.co.uk You should put "London Marylebone" as the starting station & "Leamington Spa" as the destination (this will give you a direct train) but you can also travel from "London Euston" but that's not direct.
The taxi will probably be a little more than a couple of quid, but expect to pay nomore than about £6 - £8 pounds.
The station in London is located here:
http://www.streetmap.co.uk/newmap.srf?x=527578&y=182057&z=1&sv=527750,182250&st=4&ar=Y&mapp=newmap.srf&searchp=newsearch.srf
just along from Baker St, Madam Tusaudes and a host of other tourist trap attractions.
Note however that Marylebone is only 20 mins walk from the Wallace Collection (http://www.streetmap.co.uk/newmap.srf?x=528249&y=181554&z=1&sv=528250,181750&st=4&ar=Y&mapp=newmap.srf&searchp=newsearch.srf).
which a great (free) musesum of historic armours (along with some rather chintzy 18thC collections too!).