This would cost £83.70 for an off-peak return to BIF and £38.50 for an anytime single Sunderland to Leeds (no railcard discounts included, you may be able to get this cheaper by splitting/booking advance). So you might well be best doing as John suggests if you can
Sunderland to Leeds is cheaper by splitting.
Barclay - Are you getting the Grand Central? This is the best way to go, especially with a bike as they can put it in the power car and you can sit next to it in either standard or first class (I am pretty sure they can use either power car). I have no idea about Adelantes as I've never got on one with a bike.
Sunderland - Hartlepool £5.40 (Anytime Day Single, Any Permitted)
Hartlepool - Sherburn-in-Elmet £10.90 (Off Peak Day Single, Not Darlington)
Cycle Sherburn to South Milford (about 1 mile)
South Milford - Leeds £4.40 (Anytime Single, Any Permitted)
TOTAL = £20.70
Or, if no suitable train to Sherburn, replace South Milford to Leeds with Church Fenton to Leeds for £6.50. If you want a fast train to Leeds, then it's £10.40 for a Off Peak single.
If you are using Grand Central, which I recommend, then split at York only:
Sunderland - York Anytime Single £11.00 (Anytime Single, GC Only)
York - Leeds £10.40 (Off Peak Day Single, Any Permitted)
Total - £21.40
As for the cycle ride, I take it you are going via Parkhead, Waskerley and Consett? A group of us on this forum have done this ride a few times, from Stanhope to Newcastle / Durham / towards Sunderland (the routes diverging at Consett), although we didn't go all the way to Sunderland (we actually got to just before Annfield Plain when we realised we were going to miss one of the few trains from Chester le Street so we diverted to Lanchester and took the route to Durham insted. We are aiming to go again some time. Which route are you taking beyond Chester-le-Street / Washington? The route seems pretty obvious until
here where there seems to be a couple of alternatives, one north of the river and one via South Hylton.