Good on you for wanting to turn things around. Please don't worry about losing a specific amount of weight by a particular date. What you want to do is be healthy. Once you've been exercising each day and eating healthily each day for only a few days your confidence will zoom and then who cares what you look like in a swimsuit? You can get out there and get your beautiful jiggly bits jiggling.
Scientists are the go I reckon, and I pay taxes, so I give credence to my government's recommendations for eating healthily and exercising regularly. You are very much on the right track: ditch all the processed and junk food and get cooking. Five different coloured vegetables and two different coloured pieces of fruit as your base line each day is a good lift off.
Really does sound like you know what to do, so it could be worthwhile to sit down and actually write down what is blocking you, and come up with a few strategies to get you over those hurdles. For example, you mentioned you are short of time and don't really like cooking. So maybe on the weekend when you have some spare time do some research on healthy and easy versions of your favourite meals, make sure you have the ingredients, and think ahead about what you are going to cook and eat when you are busy. It might suit you to purchase a slow cooker, or to cook on weekends stuff you can freeze. Put some real thought into planning and you will save heaps of time and end up with easy to cook healthy food you love. Maybe sign up for a cooking class.
Hit the book shelves. Michelle Bridges, for example, has cookbooksthat are super easy to follow, quick to make, and comply with Australian government guidelines regarding nutrition. Check out your local library. If you are wealthy you might want to sign up for her twelve week health kick start thingo, but a library cookbook and recipes online will get you there if you are broke.
Research shows accountability really helps, so if you can afford it consider a gadget to help keep you on track, like a fitbit. There are free online diaries such as MyFitnessPalthat are great.
Once you start eating healthily and exercising regularly you'll find it is like a drug. There is a reason all that research is recommending walking for mild depression. Eating well and keeping moving can really make you feel better and give you more energy. The extra happiness and energy, and confidence, will all be visible. By the time you get to Mexico you will be blooming with good health.