When you say your heart rate goes above the recommended max are you referring to the reading taken from the cross trainer? If so, ignore it as I've been told these are rarely accurate, so don't be put off!
When you do your cardio the most effective way to train is to do 'intervals' so if you are on the cross trainer, try doing 3 minutes on level 1 at a steady pace and then try putting it up to level 3, even if it's just for 30 seconds, then drop it back down to level 1 to recover for three minutes, then back up. You can do the same on the bikes, or alternate between walking and jogging on the treadmill. Try and aim for 15 minutes of this to start with.
When you do your weights, the mistake lots of people seem to make is to rest for long between sets, so your heart rate drops. Try to alternate between an upper body exercise and a lower body exercise so then one muscle group gets a rest whilst you work the other. For example you could do ten squats alternated with ten push ups and repeat 2 or 3 times.
Make sure you are using heavy enough weights, if you can lift it comfortably more than 13 times then you should increase the weight.
Make sure you have a good stretch at the end.
I know you were looking for reassurance that it will get easier, but to be honest it SHOULDN'T get easier as you should always be working as hard as you can. Hopefully though, you will gain in confidence and will begin to enjoy it
It can be a good idea to keep a training log, just take a notebook with you so you can write down what weight you used on each exercise and how many reps you manage, this makes it easier to make sure you are lifting the right weights on each exercise and is also really motivating as you will be able to look back in a month and see just how far you have come.
Make sure you ask the gym instructors if you have any questions, especially if you are unsure of how to do certain exercises etc as you don't want to injure yourself with poor form.
Good luck, I hope you start to enjoy the challenge.