I'm actually more of a MAN regular but living in Stoke I also get down the BHX frequently.
Firstly, BHX's runway (33/15) basically runs from SSE to NNW so it's always useful to check the wind direction on the BBC weather site or similar (Metcheck.com is good also). Once you've established runway use, there are several good photography spots, all of which can be reached by either train or walking...
1. Long-stay car-park
Exit the terminals and head right until you get to the car-park entrance, then walk all the way to the bottom, beyond the fuel silos. Here you will find a great view of the taxiway for 33. A small set of steps is useful but not essential as there are a few holes in the fence. This spot is literally 10 minutes from the terminals and is good from sunrise until around mid-day.
Example image: http://www.airliners.net/photo/Bmibaby/Boeing-737-59D/1640719/M/
2. Embankment by Total garage
Continue past the long-stay and follow the footpath to the A45 dual carriageway. Continue along the carriageway (on the right side) until you come to an embankment more-or-less opposite the Total garage. A good spot early morning for arrivals on 33.
3. Field off A45
Cross the A45 from the footpath and walk past the Clock Inn and Total garage (left side). Just after you've walked under the approach there's a metal gate leading to a field off to the right. The back of the field is good for 33 approach shots in the afternoon/evening.
4. TriStar Hotel
On the same side as the embankment but 10 minutes further up. Just beyond the hotel is a rough field with entry via holes in the hedge. Line-ups on 33 can be had here but a relatively tall ladder is required - otherwise heavies departing 15 can be photographed in the climb-out. This spot good in the afternoon/evening.
5. Marston Green train station and Sheldon Country Park
The railway bridge at the station is good early morning for taxiing and rolling aircraft using 15. When the sun moves round later, head into the park and follow the western perimeter for some good spots. A few of them require a ladder. There are many decent spots in the park so finding somewhere to suit won't be hard.
Additonally there are a few reasonable spots around the Elmdon area but without transport getting there will be difficult.
There's no reason your first visit to BHX should be a 'recon' mission - there are enough easily-accessible spots to make a first visit very successful.