If you want to use a taxi to get around, it is convenient to use the Uber app to call a taxi.
In particular, if you want to call a taxi using the Uber app at Birmingham Airport, there is a Car Park 4 Drop zone, wheich you do not need to pay any paring fee to call a taxi.
Even if you don't have the Uber app, there are plenty of taxis waiting at the airport, so you can easily catch a taxi.
A taxi to the University of Warwick will cost between 20 GBP and 30 GBP depending on traffic.