I wouldn't listen to your friend too much, "furthermore" is a classic substitute for 'also' and I as a native British English speaker use it all the time in formal and informal contexts to replace the word 'also'.
You can also use 'in addition to', 'as well as', 'likewise', 'moreover' to achieve the desired effect.
This is a problem that is quite common to be honest and you will have to repeat the word also at some point so don't worry too much.
You can also use words like 'secondly' and 'finally' to replace 'also', but obviously only in certain contexts.
Note though that you could have a sentence such as 'I went to the shop and also went to the park", in this and similar cases the direct substitution for 'also' doesn't work (maybe this is what your friend was talking about).