I had a go at doing some research a couple of years or so ago, but didn't get very far. On my mother's side, it's all a bit messy. She was adopted but had some info on her birth family. My father did some research on her possible father (an unusual last name helped with that) and she subsequently met her putative siblings. But I wasn't invited to those meetings and was told very little about them.
But around that time, things were such that I cut contact with my mother, and hence the rest of the family, so I know little more about that side of the family, except that her birth mother married her putative birth father about three months after the birth, but never reclaimed her daughter/my mother, and went on to have four further children in as many years, but apparently left the family when they were still quite young. I haven't been able to find any trace of her.
But, prompted by this thread, I've restarted the little research I had done on my father's family. There is a similar kind of problem there, in that we lived in my mother's home town, hundreds of miles from his family, and there wasn't much contact - this was back in the 1950s, so home telephones weren't particularly common.
I have so far found who his sisters were married to and the names of their possible children (my cousins), some of which I vaguely recognise.
I have ordered my paternal grandparents' marriage certificate, though I had to choose which spelling of his last name to use - hopefully the GRO will be able to find the entry because I had included the volume and page number for both. I have found the birth date of their first child as October on one site, but on another it's listed in December, which is also when his parents' marriage is listed, same year.
One other thing I've thought of doing is getting my father's WW2 army record. I'm not sure whether to try now, putting myself down as his next of kin, even though that's technically my older brother (if he is still alive - no contact with him for years and frankly wouldn't want to), or wait until next year when my father will have been deceased for 25 years which is the point where the restrictions are lifted.
One thing I have learned, however, is that I must make meticulous notes! A couple of times today I've thought I'd found some new information on one site, only to see that I had previously found it on a different site a couple of years ago.