When my dog eats grass, do I need to be worried that he will eat something poisonous, or do dogs generally know what is / isn't poisonous and stay away from poisonous plants?
Wild canine are able to smell and taste poisonous plants (or more specific some toxic components) and as such will simply avoid them as much as possible.
Domestic dogs often don't possess that ability (or it's hindered/not fully "functional").
A puppy will often just try to chew on everything and every plant (also for a learning experience), so best advice is to train your dog and don't let it unattended or off-leash until you know this is a non-issue.
It's also important to note that there are obviously different levels of toxity in different types of plants.
Keep the worst offenders out of the dog's reach (probably give them away rather than trying to determine "reach" here). For less poisonous plants (like those that might just cause some diarrhea), just some training should be good enough.
As for actual grass (when going for a walk for example), there shouldn't be too much to worry about.
Don't let your dog chew on just any plant. Common grass is perfectly safe and even healthy for their digestion (assuming there's no fertilizer or insecticides on it).
I'm often on pastures where there are many different plants in the grass and so far I've never seen our dog touching anything that wasn't a classic blade of grass (except one or two dandelions).
But besides that, I think there are bigger dangers of your dog picking up something else that's no grass or plant rather than grabbing some poisonous plant. So best advice is probably to keep your eyes open. Also don't forget about mushrooms.