Cycling is increasingly popular in Liverpool. The city is relatively compact and there aren't too many bad hills, and the fresh air from the river can make cycling a very pleasant experience (In the summer!)
Cycling along the waterfront from town to Otterspool is a lovely route, and the Woolton cycle route is also very good. Merseytravel do cycle route maps for various areas which are available at most railway and bus stations.
Risk of Theft
There are plenty are places to lock up your bike in town, however the city has some very highly skilled biked thieves operating in very organised units, so if your bike is anything other than a total piece of crap, it's best to lock up your bike in doors in possible. If you do have to lock it up outside, do it somewhere very public and make sure you use a decent D-Lock through your frame and back wheel. Something to secure your front wheel would also be wise and if your saddle is any good, remove it and take it with you. Believe me, these bike thieves are ruthless (Though London is much worse!)