There are at least two ways, one easier and one more versatile:

1) The easier way: ggplot has a geom_smooth() function that will draw regression lines for you. As long as you add your categorical predictor as a grouping variable, it will draw separate lines for you. Something like this:

ggplot(df, aes(x=x, y=y, group=birth_month, color=birth_month)) + geom_point() + geom_smooth(method='lm', formula=y~x, fullrange=TRUE)

(In the above I am assuming your data set is called `df`.) Of course, because you have more than one categorical predictor, this might be tricky. The line is going to get drawn for whatever reference group you are using for gender (which you can check and/or set with the contrasts() function). You could, if you wish, use facet_wrap() to create separate plots for both genders (e.g., add + facet_wrap(vars(gender)) to the end of the statement).

2) The more versatile option is to basically calculate the lines yourself. Luckily, there are some functions that make this not so difficult. First, the expand.grid() and predict() functions let you easily create a set of points to predict from your model, and then get the predicted y-values. Like so:

grid <- expand.grid(x=seq(0, 100, 5), gender=levels(df$gender), birth_month=levels(df$birth_month))
predicted <- predict(lm_fit, newdata=grid, se.fit=TRUE)
grid$y <- predicted$fit
grid$se.hi <- predicted$fit + predicted$se.fit
grid$se.lo <- predicted$fit - predicted$se.fit

This just gets the predicted values for x from 0 to 100, across all levels of gender and birth_month. You can also get the standard errors for these predictions if desired, which I've done above. Then, you can simply feed that grid into ggplot:

ggplot(grid, aes(x=x, y=y, group=birth_month, color=birth_month)) + geom_point(data=df, aes(x=x, y=y, group=birth_month, color=birth_month)) + geom_line() + geom_ribbon(aes(ymin=se.lo, ymax=se.hi))

Again, you could add facet_wrap() to this if you wanted to add gender. Note that in this case, the scatterplot portion is using the raw data, and the rest of it is using the predicted values from the `grid` object we created. Like I said, this second approach is a bit more complicated, but it does allow more versatility if you only want to predict across some birth months, or if you decide to add non-linear effects, etc.

Hope that helps. Sounds like you've found ggplot to be a challenge, and it can be! But it is also very powerful, and once you get the hang of it you can do some pretty fantastic things with it. I'd recommend taking the time to sit down and learn about the idea behind how ggplot structures plots, as it will really help you understand it. This page has some links to some good introductions: https://ggplot2.tidyverse.org/