Yes, welcome to this board! From your name identification I see you guys are from the Netherlands so it's a small world because our youngest son's girlfriend is also from the Netherlands (Utrecht, to be precise) and is working on her doctorate here in the states with our son. You could say they are 'domestic partners.' You've come to the right place with your questions, because the people on this board love St. Maarten and are exceptionally knowledgeable about the island. All of us were 'first timers' at one point, and it really helps when you ask questions and obtain advice before making your travel plans.
Carol and Scubaman above are both correct when they advise to rent a car. It gives you greater freedom to come and go as you please. In the long run it will be cheaper than taking taxi's everywhere, but if you have reservations about driving after dark a taxi is always an option to consider. Keep in mind that if you have a Garmin GPS device you can download maps of St. Maarten, but in reality it's hard to get lost here (though I have done it once or twice). If you go with a rental car, remember to always keep it locked and never leave any valuables inside -- ever. After you land at the airport, there will be many local taxi drivers lurking around outside to take you where you need to go. So, you should have no problems securing one quickly to take you to the french side and LaPlaya. Or, you can rent a car right after you land (there are no shortage of car rental companies flanking the airport). One suggestion that I can give you is checking with Kennyscarrental before you arrive and making arrangements with them. They participate on this board and have provided good service to many visitors in the past. They can meet you right at the airport so I would suggest checking out their website and Facebook page.
Once you have your rental car, the drive to French St. Martin will take around 45 minutes or so depending upon the time of the day. Traffic can and will get heavy, but I always try to sit back and enjoy the ride because I'm on vacation!
As Carol mentioned, there are many fine restaurants in and around Orient Village. Some examples include Le Piment, L'Astrolabe, Yellow Sub, and so many more within walking distance. You can even opt to eat at a beach restaurant such as Bikini Beach or KKO right on Orient Beach (but, pay attention to there closing times). There's even a clothing optional open-air restaurant that serves lunch called the Perch Bar & Grille on the south side of Orient Beach which serves good burgers. If you've never eaten lunch naked before, this is the time and place to try it!
Having a rental car will also allow you the freedom and drive to Grand Case (just north of Orient Bay). There are several fine restaurants here including Bistrot Caraibes and Ocean 82 which re-opened recently. I typically pay with euros on the french side, unless a restaurant happens to accept U.S. dollars at the same exchange rate. Then, you are better off paying in USD. Best to ask before assuming this. You also can drive back to the Dutch side and there are some nice restaurants around Philipsburg including Mark's Place, the Ocean Lounge (at Holland House hotel), and Chesterfield's (near the cruise port). At these places you can simply pay in US dollars, but you'll also have the option to pay with the Netherlands Antillean Guilder for which many restaurants post prices in both currencies. Bear in mind that one U.S. dollar is about 1.69 guilders so once again it pays to do the math if you plan to carry both currencies with you. And, as Carol warned watch the mandatory service charge! It's kind of like a built-in gratuity so no need to overtip!
As for must do's, definitely visit both Maho and Mullet Bay beaches on the Dutch St. Maarten. On Maho the planes land right on top of you so that's something to experience on SXM.
Keep in mind that no questions are bad questions. People like Carol and Scubaman (and myself) know a great deal about St. Maarten, but we're still learning new stuff about this place all the time. Hope you have a great time.
