Train would be your best option - there is a train station right next to the airport with a free magnetic rail link to it - you don't have to go outside.
See www.nationalrail.co.uk for times of trains etc . It should only be about an hour and a half journey max.
Bus probably takes at least twice as long and you may have to go into Central Birmingham to catch it
With the train as irishlinks observes Birmingham International station is right beside the airport (it's shared between the airport and the National Exhibition Centre) ... The free Maglev train from the terminal to the station hasn't worked in years though I think ... you can walk it or take a free shuttle bus now I believe.
Get the train! I lived in that area for a while and when the trains were down it was a nightmare to have to use the bus serivce. It was slow and very uncomfortable, not ideal at all. The Virgin Trains from B'Ham to Manchester are superb, very comfortable and nice way to travel.
. The free Maglev train from the terminal to the station hasn't worked in years though I think ... you can walk it or take a free shuttle bus now I believe.
Indeed you are correct my info was out of date (it's clearly been a while since I've flown to Birmingham and gone to the train station ) There is indeed a shuttle train but in my defence it's correct to say it's not the MAGLEV train that used to be there (which they got rid of in 1995)
Details on www.bhx.co.uk which is the official Airport site