In discord, can my friends see what servers I have joined?

Friends can only see servers that you are a part of with them. If you join servers that they currently aren't in, they won't see that you are a part of that server. That can change if they decide to join that server, if they do join that server, then they can see that you joined that server. If the friend you're speaking of leaves a server you are in, they won't be able to see that you are in that server.