Using pen and paper is definitely a good way to start off your family tree quest. But so is using a computer of sorts. Maybe what is needed is some sort of compromise and it just so happens I might have one! Using some sort of word processor, create a "family template".
In the top half of the sheet record details of the father and mother of the family group in question; include space for their parents names, marriage details and whether either (or both) remarried. Then, in the bottom half, record all the details of the children: date of birth and where born; who, when and where they married (assuming they did); and (sad, but it does happen) when they died if unmarried. (If there is census info available this too could be recorded in the bottom half!) For the children that married, a further template can be filled out, as can one for all the parents, brothers, sisters, etc of the original family group.
Using this type of template (which of course can be adapted to suit and doesn't even have to be created on a computer) means that family details are stored in a way that is easy to follow and allows simple cross referencing (especially when used alongside a more traditional family tree type layout). Its certainly worked for me on a number of family trees that I have traced and/or helped on!