Me and my wife are looking to start cycling in the nearby nature. We aren't expert biker cyclers but are in reasonable shape (we're in our 20s). We're looking for two entry level bikes, each up to the price of 350 USD. The nearby nature includes mostly 4X4 paths, sometime narrower but in general in such a paved manner. Could anyone recommend good bicycles in this price range that will fit our requirements?
First of all, why do you want a mountain bike?:
The nearby nature includes mostly 4X4 paths, sometime narrower but in general in such a paved manner.
This is definitely not what a mountain bike is for: They are intended for serious trail use, i.e. broken paths with lots of obstacles/debris. Given what you're describing and your beginning riding "style", I'd personally recommend either a lower-end touring or hybrid bike with tires more meant for off-road use: They will be more comfortable to ride for longer periods, will be easier/more enjoyable to ride fast (less rolling resistance) and can be easily used for stuff other than trail riding such as hauling groceries with a rack and panniers. Mountain bikes are a pain to use for anything other than, well, mountains†.
However, in order to get a new bike which you will really enjoy riding and can depend on for a long period of time without it breaking, you'll unfortunately have to spend a lot more than $350 USD (cf. a 2015 ranking of good entry-level mountain bikes, on which most start at around $500). Unfortunately, although bikes at e.g. Walmart may be in that price range, you will regret spending that much when just a few hundred more will get you a very respectable machine.
However, you can find a great deal on a used mid-range bike on sites such as craigslist: By buying used, you save the instant "drive it off the lot" depreciation which comes with a brand-new machine. If you don't know anyone who can help you find a good deal on a used bike, however (it's like buying a used car: be careful and always be ready to pay for repairs after buying it), many reputable LBSs will have a few used bikes for sale or will know someone who does.
† Although this recommendation is biased since this is my absolute favorite type of bike to ride, if you really are into doing mixed light off-road stuff like you describe, consider getting a so-called "adventure bike", which is basically a road bike for people who aren't afraid of getting dirty. These bikes are ridiculously fun to ride on nearly any surface and you can still carry groceries with them when needed.
First of all, mountain bike riding is a great thing for a couple to get into. A great entry level bike for around $350 is a Giant Revel. It is considered an entry level bike because it doesn't have disk brakes and only has front suspension, which is all you will need for your kind of riding. I started out with this bike and had no problems with beginner riding. I later upgraded but I feel like this bike would be the best fit for you guys as of now.