how you want to do rivers really depends upon what level of detail you want to mimic. Currently you have a nicely stylized map that could either head down the realistic road or a more stylized path. Rivers are the element that will determine that.

Water takes the singular easiest path from the uplands to some body of water. That is why there are a great many cries to the heavens and gnashes of teeth when people draw rivers that split and join back and other such stuff. Often settlements pop up around water features, so that might help determine riverine placement as well. Anyway... give it a go and have some fun.

Welcome, by the way...