Do send the shower invitation or wedding invitations first?

Shower invitation first, as the Shower will precede the wedding.

AlternativelySend the wedding invitations first. That date is hard won, set in stone and guests need as much time as possible to organize their lives around it. Agreed, the shower does precede the wedding but it is a less formal, smaller, more intimate, often house or restaurant based affair that can be organized virtually at the drop of a hat. Unless, as in Europe, you change countries and really rip it up. Even so the best showers are pretty much unscripted. Normally the Chief bridesmaid / Matron of Honor would be the one to organize the 'girls' and be responsible for any invitations, keeping the bride out of the loop (for obvious reasons) as much as possible. It often works out that a quick 'ring around' to synchronize diaries is the only invitation necessary.