For the United Kingdom, my starting point is the three Free UK Genealogy sites:
FreeBMD This site has free indexes to the births, deaths and marriages for England and Wales from since the beginning of civil registration in 1837. I use this site for marriages. I sometimes use this for births and deaths, if I want to narrow down the year of the event. I generally use the GRO Online Indexes for births and deaths, as for births they include mother’s maiden name, and for deaths they include the age of the deceased, which aren’t available on the earlier indexes on FreeBMD. I can also order any certificates I need at the same time. You will need a login to access the indexes, but it’s free to create a login and to search the indexes.
FreeCen This site has transcriptions of the England, Wales and Scotland censuses. Since they don’t have all the censuses, I also use FamilySearch to find any census records that I can’t find on FreeCen.
FreeReg. This site has transcriptions of parish registers for England, Scotland, Wales and Ireland. I also use FamilySearch to find any that FreeReg don’t have, and to check whether FamilySearch has images of the parish registers.