Seriously, here goes...
I'm pretty new to the girls' world, especially when dates are involved (I just had a few ones, but it never lasted).
I began talking to a girl a week ago, and tonight we had a lovely first date (it was actually the first time we met).
It a bit scares me as I'm not used to feeling what I'm feeling now (which is what you're feeling after you had a good time with the other gender).
I like her character and her beauty.
Where do I go from here?
I'll lay it to you simply as straight forward.
Never think that just because you're new to it then that's a liability, everyone has to start somewhere...
Be very comfortable with where you're at, and never pretend to be something you're not.
Don't worry about fear and don't let it get to you, (we have nothing to fear but fear itself), think about it what's the worst thing she can do, give you a thai kick and elbow you???
You're a tough guy if you train in Muay Thai so don't forget that.
Don't be afraid to tell her you like her character and beauty, just don't be sappy or chicken about it and keep it short, compliments are best kept short especially early on. Tell her matter of fact-ly like this is how it is, look her in the eye while you say it too. Just don't rub it in. Best time is when she says something that you like, she makes a joke or says something important, tell her with a smile and confidently, (you're funny, and I don't just mean how you look...) you know throw some very light sarcasm in. Have fun with her but not at her expense.
Would asking her out for the day after tomorrow be going too fast?
Take it slow, waiting a few days to a week is best, it shows her that you have other important things in your life and don't lose perspective of that, nothing turns women off faster than a guy who seems desperate or has nothing else going on in his life so make sure she understands that you do.
There's also the thing of being afraid of disappointing her in the second date after a wonderful first one.
Don't worry about impressing her or disappointing her, just look to have a good time for the both of you.
If she's the type of girl that wants to be impressed then move on. I'm sure she is more concerned with getting to know you and having fun than being impressed.
You shouldn't be going out with her to impress her, but to see what kind of person she is and to have a good time with her.
Your whole attitude should revolve around this, not impressing her. The best way to impress her is to be relaxed and have a good time without trying to be phony.
She had a good time with you once so don't worry, she'll have a good time again.
We mostly talked during the first date over a coffee and a walk in the shopping center, and I'm not sure that just talking would be ok again.
What do you think?
The best thing is to do an activity, that always helps to keep things fresh and helps to keep the evening going smoother.
Some ideas, mini putt golf, bowling, playing pool, dancing, salsa dancing lessons, take her to the gym, hold the pads for her, a picnic near the water and play some frizbee or throw around a ball.
Just remember to gaze into her eyes a bit, smile and touch her once and a while, most importantly be at ease, the more nervous you are will only make her nervous as well, when you are calm she'll feel the same. Remember it's easier than sparring in Muay Thai, and if you're doing a physical activity then this will help keep you from being nervous.
As for the phone, just keep it short and you should be fine, tell her you prefer talking in person and aren't much of a phone person.
A very important tip I can give you is to NEVER be afraid of your short comings, everyone has them, (myself included) the more aware and relaxed you are about them the better off you are. Just realize no one is perfect and any girl who expects you to be doesn't deserve your time.
Good luck and have fun.