Sorry I was just made aware of this thread. I wasn't ignoring you I swear!
From what you have said, I agree that you would most benefit with a hybrid bike, which is what is what Guinness said of Dual Sport. That's just what Trek calls their hybrid bikes. They are mostly aluminum frames with a suspension fork. The stock tires won't be super fat or super thin, but on the fatter side. They make them in 26" rims and 700s (sometimes call them 29ers). I would recommend the 26" for you. Also they will have frame sizes, from 14" up to 18" for the most part for ladies. They have upright riding position and riser bars (straight bars). I would demo one at your local shop and start out with the smaller ones. Every company will have different geometry (specs) so one size in one company can have a totally different fit than another (just like clothes).
Now these bikes are not the same a Cyclocross bikes, or Cross bikes. Those are more like road bikes but with thicker tires, and those types of bikes are for races during the rainy season where they ride on road, dirt/mud, hop off their bike and carry it over obstacles, etc. These bikes do not have suspension. These are with drop bars (road bike style bars) and more aggressive riding position (more leaned over).
A full on mountain bike can have front suspension only (hardtail) or full suspension. These are more for downhill riding, which is not the same as trail riding. Downhill riding is like what chachi mentioned. Avoiding tree stumps, jumps, steep downhill trails. These bikes also tend to be much heavier. You can ride a MTN bike on the road, but the fat knobby tires have a lot of drag and you will have to use more effort to pedal.
If you are talking about trail, like hiking trail type, with dirt, small pebbles, and a clear path, then you don't need a full mountain bike, a hybrid will be fine.
I should mention that riding on trails or along the river bed (if you area has that) or bikes lanes at the beach is much safer than riding on the street, bc obviously the cars.
Also the prices Alisha mentioned are about right for a decent entry level bike. You don't want something too cheap bc it will feel cheap and make you not want to ride your bike. Also the advice about getting an old model is great too. Bigger shops can probably do the big discounts. I wouldn't try to get something half off from a mom and pop shop. I doubt they'd even be able to sell it for that. Markup on bikes from the big names are barely 15-20%. So to sell something at 50% off is a significant loss to the dealer.
However, shops are able to get lists of inventory of what a certain manufacturer still had in stock from older models, so it's worth it to ask what is available. Even if it's not in the shop, they can look up what's available to special order for you, and these are closeout prices, so the shop buys is for less than normal, and hopefully pass on the savings to you. Some shops will charge a fee for special orders, or one piece special order shipping, so make sure you ask about that too. Those little fees can add up to another $50 or more.