You use 'I also like beer' when you're comparing yourself to someone. Using 'Also' here is like 'Me too'
It is the same with 'I like beer too' Somebody likes it and you like it 'too'
Both sentences are correct.
'I like beer as well as wine' means I like beer AND wine
However,if you said 'I like beer as much as wine It meas that you like both of them equally.
'Other peope like beer, and so do I' is also correct but you can replace 'Other' with [I]'Some' [I]ie: some people like beer, so do I.
I think that using the word 'other' means that there is comparison between two groups of people. ie: Some people don't like beer, others like it....or 'Some people like beer, other people like beer too'