Both FindMyPast and Ancestry cost money for full subscriptions.
The Ancestry.com access (which is at no cost to you) at your local public library is most likely just access to the U.K. based info. (correction - just noticed your question is from the UK)
FindMyPast is a United Kingdom (British Isles) based database.
If you wish to find some United Kingdom based info on your family lines, then the free ones I recommend are (in no particular order) - registration is free for all:
* Rootsweb.com http://www.rootsweb.com
* FamilySearch.org http://beta.familysearch.org
* Genealogy.com http://www.genealogy.com
* Rootschat.com http://www.rootschat.com (British Isles)
Rootsweb.com, Genealogy.com, and Rootschat.com all have message boards/forums for asking family history research questions which others may be able to help you in answering. The 1st 2 listed have message boards by surname and by location (where an ancestor lived.) Rootschat.com has forums divided by location, altho' one can search the forum questions by surname. Getting answers to your posted questions may take time. For Rootschat.com, read the info for those just starting out and find out how to set your questions for you to be notified by email when someone answers.
Added: Also Rootsweb.com hosts a free UK census database (1841-1891 census records):
http://freecen.rootsweb.com
Depending upon where in the British Isles your ancestors originated, these national archives sites can prove useful, too:
* The National Archives of Ireland http://www.nationalarchives.ie/
* The National Archives - United Kingdom http://www.nationalarchives.gov.uk/
* The National Archives of Scotland http://www.nas.gov.uk/doingResearch/remotely.asp
Plus, your local public library (maybe just the headquarters branch) has great print (yah, books) resources to help with researching ancestors from the British Isles. Ask a reference librarian at the library about that.
If all this doesn't make your eyes rattle in your head :-) I think the info I've given you above will give you a good start.
Best wishes.