Try doing 3 sets of 12 reps instead of 2 sets of 20, you will only actually be doing 4 less reps in total (36 instead of 40) but it means you will be able to add weight. If the barbell is too heavy yet to do back squats with then use a dumbbell and do goblets (search youtube for demo). Really concentrate on your form, your hip crease should be level/below your knee, when you get there you will know it as you will feel your glutes and hamstrings get to work helping support your weight, rather than it all being about your quads.
Definitely give the 10kg a go for the chest press. Don't worry if you can only do 10 reps, just try and add an extra rep each time you workout. Once you can complete 2 sets of 12 then add an extra set.
Don't shirk you back! It is actually one of the first places my dh noticed a difference when I started lifting, and it is really important to keep your body balanced.
Hatcam is definitely right about finding different ways to challenge your body but if there aren't any classes then just change up your exercises. Instead of regular lunges try them with the front or rear foot elevated, or a
For your abs if you can already hold a plank for about 90 seconds then try variations with an arm or leg (or both) raised, or your arms or feet resting on a stability ball. The cable machine can also be used for static anti rotation exercises which are great for your core, .
I wouldn't worry too much about cardio on the days that you weight train. If you are weight training properly with the right weights and rep range and rest intervals then you won't need to hop on a treadmill for five minutes in the middle of your workout to get your heart rate up... it will already be up! If you are going to do any then do it right at the end after you have finished all your sets and reps, and fifteen minutes of intervals will be more than enough. Or you could try Alwyn Cosgroves evil body matrix finisher: 24 squats, 12 lunges on each leg, 24 jumping lunges, 24 jumping squats. Rest for as long as it takes you to complete and then repeat... then collapse on the gym floor.
Hatcam can probably advise more on cardio than I can. From what I understand I think that long steady state cardio tends to make your body start burning muscle for fuel after a while rather than glycogen stores/fat whereas intervals when done properly make your body work in the same way as weight training does, ie intense effort for short periods of time with brief rest periods in between.
Sorry to have waffled on! One more tip is to make sure you have a good high protein snack with some good carbs as soon as you can after your workout as this is when your body really needs it to refuel and rebuild and it will help you see the results more quickly.
Whatever you do make sure you don't try and overload and do too much in a week. You need to always leave at least one day in between weight lifting sessions as your muscles need time to recover or you could actually end up weaker than when you started! Don't feel like you have to be doing something long and intense everyday, and don't be scared to take days off to give your body a rest. Most of all, enjoy!