I'm new to weights, but I started this week and have a few questions.

Am I doing it right if I'm getting out of breath in my workouts? It seems like a cardio workout as well as weight training to me.

Am I doing something wrong if my breathlessness is stopping me from moving through each set quickly without resting between each exercise?

Am I doing something wrong if I'm unable to do an exercise for a particular muscle group due to an untargeted muscle being fatigued? (My example here is doing bent-over barbell rows, I am unable to do them properly because my legs are so shaky from doing squats and lunges).
Perhaps I should do them stood up straight instead?

Finally, should I do reps until failure for every exercise in every set? Because at the moment I only achieve failure in the last set of my workout. And does this mean I should rest a while between exercises and sets so that I get failure at the same point each time (rather than the reps decreasing)?

I hope that isn't too much of a question overload.

Sounds like you're doing a lot of full-body exercise. I recommend working different muscle groups on different days so you can give past muscle groups time to rest. Remember:

Muscles grow at rest, not while you work out.

If I ever combine leg exercises with other workouts, I always do them last, since they're very important to a number of other exercises and you want them at the top of their game. I once made the mistake of working my legs before my cardio -- I couldn't run.

You should altenate exercises so the muscle groups fatigue together. E.g., do a set of triceps, do a set of biceps, a set of triceps, a set of biceps, etc.

At the end of a set, yes, you should have reached muscle failure. Try to find that sweet spot between weight and reps.

If you want to grow muscle faster and strength not as quickly, try to reach muscle failure in 20 - 30 reps with very little rest in between.

If you want to gain real strength but not necessarily increase size as quickly, try to reach muscle failure in 6 - 12 reps with plenty of rest in between.

I have a different take on your breathlessness. I had the same thing happen when I first started doing full body workouts. I udjusted my rest time inbetween sets and that helped quite a bit, the rest was taken care of after my body was conditioned some more.

Most guys I know work weights until technical failure, not muscle failure. Technical failure being that moment when you loose your form.

Thanks for all that help, I'll try all the things you've suggested.
About the "different muscle groups on different days" thing, I kinda already am. I'm following a workout that ManofSteel gave me which is as follows:

Day 1
Lunges 5x5
Rumanian Deadlift 5x5
Bench Press 5x5
Barbell Row 3x5
(Squats 5x5)

Day 2
Deadlift 5x5
Military Press 5x5
Chin Up 5x5
Barbell Curl 5x5
(Dumbell Flys 5x5)

(I added Squats to Day 1 and Dumbell Flys to Day 2, and will probably add Skullcrushers to Day 2 as well). I have 2 days of rest between each workout day.

ManofSteel told me that this workout will give me all the size and strength gains I want (even though I'd rather focus on size). So if 20 - 30 rep range is the best for size then does that mean ManofSteel is wrong?
What are the hard facts about rep numbers for different purposes?, because it's beginning to annoy me when I keep seeing different people saying different things.

Now that I have completed both of these workouts for the first time, I noticed that the the 1st one kills my legs, whereas the 2nd kills my arms. Does this mean that the muscle groups are split fairly well?