Assuming you want a reasonably direct route, the A583 isn't too bad despite being a major road - it used to have two lanes each way but was reduced to one lane, with a cycle lane separated by wide road markings, and a 50mph limit.
The trouble is, getting to the A583 can be a bit troublesome, the main problem being the junction with the M55, which has no cycle provision, and southbound the left two lanes sweep onto the motorway without a give-way; not a nice place to be on a bike. This is one place I'd give up my right to the road and walk if need be - it's treated by drivers as if it's part of the motorway and is a classic example of barriers to cycling. I think there ARE footpaths and a pathetic attempt at shared use, but they're on the south side (the shared use dumps you into the road just before the roundabout!).
My suggestion - though bear in mind I know these roads from weekend leisure rides - would be to head along East Park Drive, Lawson Road, Lancaster Road and Preston Old Road, join Preston New Road, then turn up Mythop Road, through Weeton and Great Plumpton, joining the A583 a little further on. Mythop Road onwards are NSL roads and unlit but not really all that bad in my experience.
What kind of bike are you using? There's a network of bridleways between the hospital and Staining, which give access to the B-roads to the east of Blackpool without going through the urban area. One route goes round the golf course and on to Staining village, while another heads to the Marton Mere caravan site - and possibly onward to Mythop Road. The surfaces are firm but a bit rough, and I wouldn't go round there at night, but it might make for variety. The big downside for me is Chain Lane, between Staining and Mythop Road, which is a rat run with dips and bends that make cyclists vulnerable.
I'm sure someone with more direct knowledge will come along - my main tip is to avoid the A583 / M55 junction if at all possible.
