I don't like the type of fantasy that includes wizards, magic, dragons, castles, swords, etc. I like military SF, space opera, cyberpunk, and hard SF (as long as it doesn't get bogged down into the technological aspects).

But what I really like is story-driven SF - a good story with well-drawn characters and believeable dialogue, that just happens to take place on another star (planet), or in another time. A perfect example of what I'm talking about would be Clifford Simak's Way Station. What would you call this type of SF?

I don't think that kind of SF has any particular subgenre, as any of the subgenres you mention can feature good stories, characters and dialogue. I haven't read Way Station (though I've always intended to) so I don't know for sure that I can provide a good recommendation, but I think Lois McMaster Bujold writes science fiction like that. Her Vorkosigan novels are military SF or space opera, but they are definitely story and character driven and I giggle out loud every time I reread them at the dialogue.

I've read Way Station and other Simak books as well.  I can't really think of a subgenre for it; I just think of it as 50s and 60s science fiction.  You could try some other Simak books - I would especially recommend City. 

Way Station won the 1964 Hugo, so you might also like other sf classics from that era...

The Stars My Destination, by Alfred Bester

More Than Human, by Theodore Sturgeon

A Canticle for Liebowitz, by Walter Miller

Childhood's End, by Arthur C. Clarke

A Case of Conscience, by James Blish

The Moon is a Harsh Mistress, by Robert Heinlein

The Man in the High Castle, by Philip K. Dick

well, for one, fantasy is a different genre than scifi, even though they get lumped together at the bookstores and libraries.

I'm guessing since you like Simak that you'll like the oldschool scifi, like the others above have mentioned. It is all yours ;-)

The genre your looking for should be titled "Adventures in Space". check out Heinleins Cat Who Walks Thru Walls or the Moon is a Harsh mistress, not to technical but a load of fun!