How to store cut fruits & vegetables for days?
Both methods work. If I had to pick one, I'd pick frozen vegetables. If you can afford to go to the store every day and only eat things that are in season, buying fresh will get you more nutrients. But things frozen when they are perfectly ripe retain most of their nutrients so you're really not losing much for the convenience. Time is the enemy of nutrients, so if you can buy local also means it's not sitting in a truck for 5 days, so the closer you can get stuff to your home, the better. Freezing is like really slowing down time. 

Lemon is mostly used to stop fruits/veg from turning brown, so especially for apples, bananas, and avacado. It will extend the life of food by up to a few days. The downside is the lemon can overpower the actual food, but you can store it in your fridge and the fruit/veg will keep most of the same texture/flavor, which is important for some dishes. This will only slightly extend the shelf life of your food and as far as I've read this is more about making the food look good. You can also do this trick with vinegar. It's the acidity that does the trick.

Freezing is great, and you can keep things for ages(sometimes a year or two) without losing nutrients.  The downside being that the texture/flavor of the food can change. But this will be fine for most dishes. You can have smoothies, fruit sauces, soups, lasagnas, casseroles, stir fry, and a bunch of other things without really noticing the difference. Some things are even fine by themselves. Sliced frozen bananas and strawberries are a great treat and healthy ice cream replacement. 

Airtight is important, as you said--ziploc bags are fine though you can also get airtight glass containers. It's a bit of an investment, but you save money in the long run by not throwing away food.