I need help when it comes to buying and preparing meals

As long as you log your food and eat at a deficit you will lose weight. Even if it is what your family has cooked. Just watch portion sizes, and try your best to log appropriately. If their foods are too high in calorie for you, try just baking chicken (I use a rub if I am cooking it in the oven) and frozen vegetables! It doesn't take much time to do that. I stock up on the frozen veggies and chicken/fish/burgers, anything I can quickly make when I get home from work. I do not really look at fat, carbs, sugars anymore. Just calories. That is what works for me.
Fruit makes me hungry! Though I still eat it because I like it. Protein is what helps my hunger.I have also learned that I do not need to eat a "traditional" dinner anymore. You know, meat, potatoes, veggies, and bread...Sometimes I will just eat the meat and veggies, or the meat and a roll. Sometimes it is a yogurt, eggs, or a protein shake. As long as I stick to my calories on most days, I am in good shape.
I love walking around the market to look for quick, easy meal ideas! Good luck to you. I hope you find what works for you!!

When I eat bananas or apples, I have to eat almost all of them because they don't feel me up

LOL...okay sorry I just had to get that off my chest, I spit milk over here when I read that. Great mental image of you being annoyed that your apple wasn't coping a feel.

If your intent is to lose weight it doesn't really matter what you eat as long as you:

A) Are eating less calories than you expend
Are getting sufficient nutrition to be healthy.

You don't need to eat only fruits and vegetables and whole foods and blah blah. You can eat what you want, just in moderation.

As for how to feel full yeah I would suggest eating more lean protein and veg as both are very filling for the amount of calories they contain.

I don't like Lean Cuisines and things like that because #1, they don't taste that great and #2, they do not fill me up. I am sure they are good in a pinch but I prefer the food I prepare. I personally wouldn't eat them on a daily basis.

You don't need to be "filled up" - you just need to get rid of the hunger pain - if it comes back in a little while - eat something else small. Try eating a banana and drinking a giant glass of water. Eat protein, moderate healthy fats and slow burning carbs. Sugar tends to leave you especially dissatisfied.

I like to cook ahead of time if I know I'm going to be very busy. I personally don't like frozen meals, because they're so overpriced. Much cheaper to mimic the recipes myself!

One recipe I like that's super nummy and kills an alfredo craving is to buy a Birds Eye Recipe Ready package of frozen veggies, use 1/4 a cup of Prego alfredo, any pasta of my choosing, and 4 ounces of chicken breast. Their alfredo is surprisingly low calories/fat (80 cal/7g per serving, 3.5g sat fat), but it's great if you need to get some extra calories in for the rest of the day.

Another one I enjoy is southwestern veggie soup. If you want the recipe, shoot me a message in my inbox! I don't use any salt in it, but it packs a helluva punch to your taste buds, and the veggies fill me up no problem.

Should I start cooking my own meals?

Yes. Either that or hang out in the kitchen and measure everything they put into the food. Otherwise it's very difficult to accurately log what you're eating.

And don't discount the value of fat, it's the most filling macro. Trans fat and partially hydrogenated fat are bad news, but saturated fat (like from meat and butter) and mono-unsaturated fat (like from olives and avocados) are good for you. You just have to work it into your calorie budget. Fruit isn't likely to fill you up, carbs generally don't. Your body burns them too fast.

I don't like cooking, so it's taken me a long time (and several failed diets) to find a rhythm that works for me. But here's what I end up doing. I keep meat and vegetables available at all times. I don't buy much fruit, but that's cause I'm on a low-carb diet. Other than that, I keep some salad dressings, and bacon and eggs handy as well. I also have a good sampling of dried spices (I don't cook often enough to make good use of fresh herbs).

When I buy meat, I buy whatever's on sale, and whatever I feel like buying. I usually buy more than I need. As soon as I get home, I separate it into 1-3 serving portions in freezer bags and put it all in the freezer.

When I buy vegetables, I focus on the kind that I can eat raw. This includes sugar snap or snow peas, cherry tomatoes, cucumbers, broccoli, cauliflower, spinach, lettuce. I always have leafy greens available, and I try to mix up with spinach, kale, romaine, etc. Focusing on vegetables that I don't have to cook makes it more convenient for me to eat them.

I take a package of meat out of the freezer every couple of days, and I'll cook 1-3 meals at a time. All I do is pair up the meat with some vegetables; maybe add some spices, oil, or salad dressing or other minor ingredients (like peppers and onions) to dress it up. I sometimes look up recipes. Sometimes I just wing it. Starting out most of my food was really bland, but I've learned how to make it better. It just takes some practice, creativity, and perseverance. The important thing is to weigh, measure and log everything. This way, not only can you look for patterns in what you eat that made you feel full, but you can also copy your meals from previous days if you're not feeling very creative.

For breakfasts, I always have some hard-boiled eggs and cooked bacon in the fridge (because I usually need something I can grab and go). I also have fixings for protein shakes if I have time to make one.

As for freezer meals, I agree with everything else people already said. But if you're not used to cooking your own meals, they can be helpful for making the transition. I mean, there will be days when you just don't feel like cooking, and you have to choose a lesser evil. The advantage of freezer meals (and packaged foods in general) is that at least you have the nutrition info so you can log it. Then, even if it's something you maybe shouldn't be eating, you can consider it in the big picture with everything else you've eaten that day.

It's tough trying to change your diet when there are other people in the house who are happy with things as they are. You mention that your family eats lots of greasy food. Is it homemade or is it processed food? Though calories are calories, if your family makes food from scratch then it means that you know your way around the kitchen. Adding a salad before the main course, and having a side vegetable (green beans, carrots, a mix of course, asparagus, etc.) that is steamed or lightly sautéed in olive oil, is a great way to start. Be consistent. If you need tutorials on how to prepare veggies, look them up on youtube.

I grew up in a household with lots of healthy homemade food, but sometimes it was very high in fat content. Luckily we always (and I mean always) started every meal with a salad, and we never ate dessert.

Start small. Add healthy sides, a salad. Eliminate beverages and have meals with water.

Also, you mention wanting to be full, that the fruits don't fill you up. Work on not eating to fullness. It takes practice, but it is the surest way to lose weight. Good luck!