If you value your fans, try not to lose by a large margin in home matches. The effect is much worse if your team doesn't score any goals while the other team scores plenty. When this happens the fans become very upset. Results away from home still have an effect, but it is not so pronounced. Conversely, scoring lots of goals, and winning at home is the best way to increase the size of your fanclub. A win in general doesn't bring in many fans. Scoring a lot of goals and winning is far better than winning 1-0 so try and play your matches accordingly.
The amount of fans will change on the daily update immediately after your home countries matchday (competition, cup, or friendly). The number of fans joining your club depends on their supporter mood (better supporter mood leads to more fans) and on fan club size with respect to division level. Each division has a soft cap on maximum size of team's fan club. The closer you are to the soft cap, the harder it becomes to get new fans.
Remember that when you lose a cup game you will take a fair hit fan wise, so don't be upset when it happens as it is expected.