Buy a ticket to Manchester at the ticket booth and go down to the platform, taking the first train to Manchester Piccadilly. There is a train about every 10 mins, and take 15 to 25 mins (some stop more). They run 4:43am till 1:22am so you are OK even if late. You can look up train times, you will need to type in from Manchester airport to Manchester Piccadilly.
Another possibility is to take a bus from Manchester Airport 'The Station' to Manchester Piccadilly train station. There are several bus lines which go this direction, for example the new X2 line which runs every hour. The bus station is just around the corner of the train station (The 'Station') at Manchester Airport. This is really just a fall-back plan in case there is something wrong with the trains. Like the stored program computer, passenger trains were invented here in Manchester. But that does not mean either work especially well!
Now Bob Paterson, who is very good at reading maps and looking things up on the internet, has pointed out to me that Manchester Oxford Road Station is closer to both Ibis hotels. It is also on the Oxford Road (as the name suggests) which is a good place to get a bus to the Alan Turing Building (if you don't want a 15 min walk) or to get a bus to Luther King House. It is not to befuddle and confuse you that I left this out earlier. Its just that (1) not all trains from the Airport go there (2) its not so easy to get a taxi from there whereas there is an enormous taxi rank at Manchester Piccadilly. But if your luggage is easily carryable or wheelable walking to either Ibis from Oxford Road Station is fine. You could even change trains at Piccadilly and get the next one to Oxford Road if you want, pretty much any train from platform 14 will do it. If you happen to fly to Liverpool John Lennon Airport and get the train it may well stop at Oxford Road anyway. One other point. There is no lift at Oxford Road so unless you are lucky with the platform you will have to haul your luggage up and down stairs.
Railway tickets saying MANCHESTER STNS are valid to both Piccadilly and Oxford Road.
Make a note of where you are staying. There are two Ibis Hotels on Charles Street and City Centre (Portland Street). There is also Luther King House which is about 2 mi (3km) from the city centre.
At the Piccadilly Station follow the signs to the taxi rank (if you have a big bag take the lift (that's elevator in American English). Both Ibises are walking distance but not with a big bag, and not if it is late at night so a taxi is probably best. For City Centre Ibis it is possible between 7 am and 7 pm, or 10am - 6pm Sunday to take the free metro shuttle from the main (upper) exit of Piccadilly station. You want the Purple route (3) route, to the third stop called Portland/Cube. In the UK the traditional style usually black taxis (technically Hackey Carriages) alway use a metre so no bartering is involved! They are the only ones you can "hail" on the street. Others called "private hire cars" you phone and they come for you.
More ideas if you fly to Liverpool. As well as getting the train, coach connection of Terravision from Liverpool airport to Manchester. The first stop in Manchester is Sackville Street (just behind the main bus station of National Express) which is VERY close to the Charles St Ibis and the second stop is Manchester city centre. Time table here. A return ticket costs just £ 8 from Liverpool to Manchester.
If you are in Luther King when first arriving I suggest you get a taxi (the address is Brighton Grove Manchester M14 5JP). When you get the hang of the local public transport and you don't have a big bag you will see that University of Manchester Campus and the City Centre are a quick and easy bus ride away from Luther King.
If you arrive on Saturday and are looking for something interesting to do in Manchester on Sunday, I recommend the Museum of Science and Industry. They have the first stored program digital computer restored there. As well as steam engines with interesting mechanical governors.
From Luther King walk west on to Wilmslow Road and cross over. Find the nearest bus stop (probably a little way south). Most busses on that side will be going towards the main University campus on Oxford Road which is the continuation of Wilmslow Road) numbers include 42,44,X57. You can pay on the bus, preferably using coins. Look out for a large grey cylindrical building (University Place Building) on the right and some fancy gothic style old university buildings on the left and get off at the nearest stop. Cross over at the pedestrian crossing and follow the pedestrian road going East just north of University Place. You will see the Alan Turing building ahead, a four storey gray zinc and glass building. Walk in between the two parts of the building and come in the double doors. Lucy and her staff will be waiting there to register you. There is a
There are further directions to MIMS/Alan Turing by car/bus/air etc on the MIMS website.