The City centre itself is not currently so well endowed with cycle infrastructure, however the main attractions in the centre are in any case best appreciated on foot. (see this review, although their Cycling strategy is ambitious.)
Outside of the centre Merseyside is more cycle friendly. Mersey Rail are very relaxed about bikes being bundled up and down their escalators and on their trains, and they offer a free fob scheme to access secure parking at many stations.
The Trans Pennine Trail passes through, whether you take the direct through route N of the city centre on the Old Loop railway line, or better still the alternative route via Sefton Park/Pier Head/the Leeds Liverpool Canal and see much more of the city.
Merseyforest publish a 32 mile ‘See Liverpool in a (longish) day?’ urban City circular ride (click on link for map) as a basis for seeing more of the city. Like the TPT, this also utilises part of the Old Loop line. (Not tried by author as yet)