"a while ago" refers to a very unspecific time in the past. When I use it, the time is usually before "recently". ("Already" is used in a different way).
When using "recently", the event is still in my conscious memory but I am not thinking of the precise time when it started. e.g. "Recently, I started learning Italian"
I use "a while ago" when someone talks to me about something they have done or are doing and I am thinking if I have done something similar.
The precise time in the past doesn't matter when you say "a while ago" but it is usually earlier than something which happened "recently".
e.g. Friend: I am going to Italy next week on holiday.
Me: Oh really! I went there a while ago too.