JunkFoodBad is completely correct. If you have a lot of fat to lose, your nutrition should be your primary concern here. How and what are you eating?
You really need to get into the habit of eating multiple meals per day; ideally 5-7. Each meal should consist of some lean protein, carbs (mainly complex except for post workout), and healthy fats. Cleaning up your diet will really make a world of difference.
What are your stats?
Regarding your questions, what kind of weight lifting equipment did you buy?
If you have a lot of fat to lose, doing cardio daily is ideal. Start slow and do it for 20 minutes. If this is easy increase the intensity, and don't go over 60 minutes.
It is a good idea to vary your cardio exercises, just to keep you motivated and to keep your body guessing. Many people buy one piece of cardio equipment and use it for a month or two then stop. Understandably, they get bored of it! So switch things up.
In regards to weight training, you can most definitely overtrain your muscles, especially if you are not used to lifting weights! It is the most common error individuals make in the gym.
I would start out with total body workouts 3 days a week. Be sure to hit all the major muscle groups: Back, Chest, Shoulders, Legs, Core. If you want, throw in a few Bicep and Tricep exercises as well. Do one to two exercises per body part, each of the 3 days that you train. Keep the workout under 60 minutes.
If you want some ideas for some routines, let me know.
Also, it is possible to gain weight when first starting to workout, it is when it is easiest to add muscle. However, it does not seem like you have been weight training long. If you were gaining weight while you were only doing cardio, I'd be willing to bet that your nutrition is off course.
Yes, you can, and should do cardio on the days that you lift weights. Ideally, you would run first thing in the AM before you eat. However, don't fret if you can't manage that.
Hope this helps!