You're working too hard, getting sore, and not wanting to continue. Normal. Back off a little - go as often, but lift less weight or do fewer repetitions. If you are using 8 pound weights, drop down to 4 or 5 pounds. If you are doing 20 reps of each exercise, drop to 12. Do something else in between so you rest those muscles before going back to do another set of reps.
Of course you're not losing weight in the beginning - you are building muscle, and that weighs more. Your body has to learn how to use that muscle to burn calories, and that doesn't happen overnight.
Be sure you are not eating too little - if you take in too few calories (especially at a higher activity level), your body will think it's starving and slow down your metabolism.
Stop listening to their promises. Sit down with one of the trainers and explain that you are sore and ill and ready to quit. They don't want that, ideally. However, understand that some of the less-reputable places make their money by selling a package that they don't have to deliver on because people get hurt.
You should be working with someone to make sure you are in exactly the right position for each exercise - the wrong position can cause injury. If they aren't taking the time with you, complain. Ask for a less boot-camp-style instructor/trainer and say you are getting ill. And do hydrate more. Don't do a bunch of chemical-and-dye laden things they are selling you though. I work in exercise nutrition, and most of the stuff they hawk at these gyms is over-processed and over-priced, and the staff is not all that trained in it. The gym managers go for the high profit margin stuff and not what really works for muscle repair/recovery.
I would not jog if you are in pain. I would walk on the treadmill or perhaps use the elliptical, but focus ONLY on time and your target heart rate, and not so much on speed or distance. And you absolutely need a day off in between - take a slow walk so you aren't need the kitchen, but stop trying to burn burn burn! Use ice to reduce inflammation in the muscles. You are trying to accomplish too much in too short a period of time, and your expectations aren't realistic. And do stop looking at the scale. The mirror will be the best judge, but not for a while. Resolve not to look at anything for the first month except whether you are walking a little farther each time.
Do you have excellent shoes? If not, go to a real running/sports store (not a mall chain) and have them watch you walk in shoes and without shoes, to look at your stride and whether you pronate, then fit you for a good shoe with good cushioning for YOUR FEET. Let them know the types of things you are doing - probably they will fit you with a good cross trainer shoe but let them guide you.