I have recently taken up yoga lessons but this question doesnt necessarily is yoga related. I found it is really relaxing when I am doing yoga exercise where I am stretching. I sort of found myself constant wanting to stretch and stay in certain positions quietly breathing while feeling that part of the body or muscles get pull it's quite enjoyable.

Is there anything I should look out for? I intuitively think it will improve the strength my endurance. My mind is so still it's like meditation.

Stretching is great fun,very enjoyable and good for you, the best stretching is done whenyou've had a warm (not hot) bath, put some loose clothing on and streeeeeeeeeeetch.
Don't bounce just stretch - whatever muscle groups, arms, legs, back and hold the position, after a few moments (when the discomfort goes) stretch further.
The main things to steer clear of is hyper-extention of joints and flexing a stretched muscle.
Hyper-extention is..... kneel down, lean back, keeping your knees close together, then place your back on the floor whilst trying to keep your knees grounded. This extends the knee joints past their optimum and can strain tendons and ligaments - DON'T DO IT. Knees and shoulders are the classic joints to injure in this way (shoulders by swinging the arms vigorously backward thus injuring the ;igaments that run across the chest).
I usually stretch whilst making a coffee or cooking, place one knee on the work surface and carry on working (keep changing legs).

Wow, Professor, either you are very tall or very flexible. I will have to do that kind of stretching on a coffee table  

I used to practice yoga for a long time, Alee. Now my laziness gets the better of me, but I still have to stretch or my back and sciatic nerve will start giving me trouble.

Keep enjoying your asanas for as long as possible! I am really ashamed of myself: should be more constant, really, but I enjoy mental exercise much more than physical and I find it hard to get time for both.  

Yes I truly enjoy it. My bad back feeling much better by doing hamstring stretches, I like warriors poses the most.

Now I truly appreciate how intelligent by  body is.

Well I'm certainly not tall - 5'4, like most things in life stretching does get better the more you do it and doing it everyday is best of all.