Good First Date place in Southwest Broward
Josef's in Plantation is very nice and has very good food and La Brochette Bistro in Cooper City is also very nice with very good food. If you're wanting to stay west, you can't go wrong with either of those. Check out their menus and decide. They are both more upscale food, but still casual.
Even though you said Southwest Broward I've still got to go with the beach. The drive over will give you time for conversation. Then take her for a stroll along the Hollywood Broadwalk and check out the menus at the cafes until something seems to strike her fancy. Find out what she likes to eat and choose something that looks good to her. (The worst thing you can do on a first date is feed her something she hates!)
Sugar Reef is good for caribbean style dishes and in my humble opinion the only real chowhound worthy place I've tried on the Broadwalk. But, on a first date the company is almost as important as the food. Heck, sharing a pizza overlooking the ocean doesn't get much more romantic!
If Thai food would be appropriate, Emerald Thai, South of 595 near Pine Island is excellent and moderately priced. If you like duck and you like mild (coconut) curries, share the Emerald Duck...very sensual! They offer excellent seafood dishes as well. The atmosphere is clean, not fancy - but it is a quiet atmosphere, good for conversation and there is no rush.
John the Baker: excellent Pizza Palour...the garlic bread and pizza are not to be missed, the other italian dishes are just so so (at least the ones I've tried). SW Pines Blvd.
La Granja: awesome peruvian rotisserie chicken. they now offer other dishes (which I have not tried) Same plaza as above (near Home Depot)
Latin American Grill: killer cuban cuisine ditto location.
Jersey Wings: looks interesting?
Then there's that slew of suburbian style chain eateries along West Pines Blvd. Take your pick!
On the Boarder (chain) is out by 75 on Pines...for poorly contrived gringo-izd Mexy.
There's an Ale House out there...among many others...
Do a drive by and pop your head in a few places.
If you're looking to stay closer to SW Broward, I've heard that Cohiba in Pembroke PInes is good. Havent' been there myself. I like Flanigan's in Weston which is very casual with good food and drinks. If you like Asian, you can try Stir Moon sushi and stirfry in Weston - very good. Lucille's in Weston is good comfort food - not too pricey.
Hmmmm....casual??? I'm not real familiar yet with that area, but I've been to Season's 52 and that has a nice atmosphere with cozy spots for a first date. I love the Canyon, but it's loud unless you're going early and during the week. How about Himmarshee's for casual semi-fusion food and then you can walk along the river. for your second date, should you bump it up a notch or two, then definitely Blue Moon cafe on the water or the River House. I'm still exploring Ft. lauderdale. And.....when you're in love!!!!! then head over to Cafe Maxx for a divine dinner and a fab bottle of vino.