Apples - they're all picked in the fall anyway and kept in cold storage. Citrus does pretty well, and so does pineapple. We cut up cantaloupe and honeydew melons into chunks and refrigerate in a container - the only time things get a little mushy is when we leave a cut half melon in with the seeds, and even then it's only the soft center around the seeds that is the problem. I don't think watermelon keeps very long. Berries keep about 4-5 days if you don't wash them first. Again, commercial berries were all picked at least a week and kept in cold storage exposed to gas to prevent rotting. I find that things from the farm stand don't last that long, and I'm kind of glad about that! I'd rather have fresh and stop more often.
I'm not sure it's practical to only go to the store every 2 weeks. Can you do your major shopping trip every 2 weeks and do a quick pick-up to restock fruit?
Frozen fruit is just as nutritious because it's frozen close to the harvest - so if you can buy that for a lot of things, you'll save money. Frozen fruit is indistinguishable from fresh in oatmeal, smoothies, parfaits, etc. My son loves to make smoothies with frozen berries, mango, melon and so on, and he throws in the fresh berries that are getting a little soft. He even throws in the green tops of the fresh strawberries he ate the day before. He also puts in frozen kale and frozen spinach.
Same goes for frozen vegetables - we eat a lot of frozen spinach and broccoli in quiches and frittatas, use frozen corn and peas in enchiladas and rice salads, frozen butternut squash in a pureed dish with garlic and onions, etc. You can use frozen green peppers & onions in tomato puree to make your own spaghetti sauce or chili - that gives you the fiber from the vegetables that you don't get in jar sauces. Canned beans are inexpensive, and dried beans are 1/3 the cost of those, so if you plan enough to soak beans overnight, you can get a lot of protein and fiber in your meals without using fresh produce.
Root vegetables keep decently - potatoes, sweet potatoes, turnips, carrots, etc. Keep them in a cool dark place. Celery keeps best if you slice off the bottom and put the stalks in cold water.