Go out and fight the new enemies that pop up till you get a skill point.
I have 40 skill points and it tells me to come back when I get more experience, do I have to wait until the next update or do I need more points?
Probably next update since everyone only goes up to 2. 3 and 4 are coming out later this year.
Fighting with anyone will raise you skill points, you will see them in the first red heart of the phone and pressing see hard numbers.
Have you already fought the rat and the horse 3 times and won?
I prefer to beat them, like that I'm sure. And even if you beat them, you can still do naughty things with them x)
Hopefully in the coming 2 or 3 months, even moar... MOAR!!!!