Women can say either of these expressions. They both mean the same thing. "Let's grab dinner" is a colloquial expression.

To me, "go to dinner" implies a slower, better quality meal at a nice restaurant. Like, "Let's go to dinner on New Year's." "Grab dinner" implies a faster, less quality affair, like going to McDonald's. ("Let's grab some dinner at the deli before the we go to the meeting!") Just my personal opinion, though.

Let's grab dinner - implies something really casual and fast. You wouldn't say "let's grab dinner" if you're going out to an expensive, sit-down restaurant. But you could say it if you wanted to go to a sandwich shop or a casual restaurant. Like "Hey, I have work in an hour. Want to grab dinner at McDonald's?"

Let's go to dinner - is just a normal expression. Doesn't really imply anything formal, but you can maybe expect to be there longer than you might be at a McDonald's.

There's no gender bias for either of those expressions ^^

Finally, we don't really say "Let's go for dinner." Instead we say "Let's go out for dinner." ^^

We also commonly say "Let's grab a bite to eat."

If someone says "I'm going to grab m-Kan's dinner" you should start worrying.