Black cabs have the monopoly and as there is a one way system in operation plus tonnes of disruption caused by the upgrading of the tram system, i would use public transport.

Option one: Walk then Bus
Come out of the exit by the really old ornate kiosk. In front of you is a huge silver building called The Urbis. Cross the road so you are on the same side as the Urbis and walk right ahead along the side of the Urbis (Toward Starbucks). When you get to Starbucks coss the road so you are on the same side of the road as the PrintWorks. Walk up this road (past Tiger Tiger) which heads toward Shude Hill. You should reach a junction where on one side you can see Shude Hill bus station and on the other side you can see the Sony Centre on the right. Go right, walk down this road passing The English Lounge, the Arndale Market etc until you see Primark. Turn left in front of Primark – you should be able to see the Piccadilly Bus Station. (5 mins)
Using the stops opposite Marks & Spencer and Quality Save get on a 142 or 143 Magic Bus. These are BIG blue buses and the fare is only £1. There are other buses but they will charge about £1.70 (give or take). The academy is on the right hand side (so sit on this side). When you see the Manchester Museum you will know yours is the next stop. (10-20 minutes tho totally depends on traffic)

Bus & Bus
Use the city link outside Victoria to get you to oxford road. This is free (beware the one way system and congestion on deansgate this is not that fast) Get off outside the BBC and get on a 142 or 143. And when you see the manchester museum it’s almost your stop.

As someone who lives and works in manchester, the more you walk the quicker you’ll move!

If you insist on a taxi, use the black cabs operated by ManTax 0161 230 3333