Hi Lynwen
They are better known as the Somerset Light Infantry. If you google "13th Foot" or "13th Regiment" you will find a lot about their history, including -
http://www.regiments.org/regiments/uk/inf/013SomLI.htm The 13th Foot had two battalions at the time you are interested in. A man enlisted in the regiment, and not into a battalion, so you have you check out both (plus the depot) as he could be transferred between all three
If you click on History on this site, then each of the battalions in turn, you will see that the 2nd Battalion were in England (which included Wales then) in 1871, and they moved to Ireland during the year, then to Scotland (1875), England (1876), and they were in India for the 1881 census. So he was in the 2nd Battalion at the time of the 1871 census. The 1st Battalion were in Ireland for the 1881 census, so that is probably why you cannot find him in that one.
To research him further you will have to visit Kew or employ a researcher. See -
http://www.nationalarchives.gov.uk/militaryhistory/army/step4.htmIf his papers exist they will be in WO97 (WO=War Office). Note if he died in service they normally threw away his papers 20 years later. Discovering if his papers survive in WO97 will take about 20 minutes, plus another 20 minutes to film them if they are there. So it need not be expensive to employ a researcher.
If they do not exist, you should find him in the Muster and Pay Lists (they are actually bound books, a lot like school registers), that show month by month where his was, his rank, his pay, when he enlisted etc. These are in WO12. Researching these can take a while but they will give you the actual town he was in while his papers will just say ‘Home’, and that would include England, Scotland, Ireland or Wales, so if you want the complete picture you should get it from WO12.
Again his papers will give his medal entitlement for fighting in any wars, but you can also check the medal rolls, which are on microfilm (in WO100) in Kew, if his papers have been destroyed.
Ken