I drifted into IT after not having the werewithal to do things like law post first degree. I started with a first line desktop job and moved into server support and then made the jump to UNIX/Linux from Windows. I did do half of an MsC in Computer Science at Birkbeck (well worth considering). I even use (abuse) some of the stuff I learn't there.
The vendor specific courses are often good but very expensive and the current climate in companies to pay for training seems to be built round the harsh laugh and FOAD principle - others can disabuse me of this idea.
One of the troubles is that the industry is in a state of change and the traditional sysadmin role is going to disappear quite soon. In my view anyway.
I would start with working to expand your knowledge by building different stuff at home. E.G. put linux on an old PC and then try to get samba and ldap working so you can get some knowledge of how these things work. You could also play with things like apache and such like in this way. Of course, this won't let you put x years experience with Linux sysadmin on your CV. Hell desk jobs tend to be poorly paid and not located in London
Good luck with the career change.
Beer because you will need it :-)