You can get to Manchester Airport from Crewe, which avoids having to go into the city and double back, provided you get a decent connection. The airport has its own station (and tram stop) so there's no need to get a taxi.

In terms of tickets, Virgin Trains offer what used to be called a "saver half" return. That means you can combine an inflexible ticket (to the airport) with a flexible one (from the airport) at a reduced cost. If you search for fares on their website, you'll be able to combine Advance with Off-Peak - that's what this is.