I just started my lesson last week (piano player trying out string!!), I am testing out two teachers, they seem very different. They are both using the Suzuki books on me though. The first one taught the basic, how to hold the violin, bow, etc, and told me to work on pizacatto on the first variation of twinkle twinkle little star, and also to work on bowing open strings concentrating on even tones.

The 2nd teacher, I still haven't took an official lesson with her, she told her to go in and she put tapes on my violin to get me started. Then she told me to go home and try to play the theme of twinkle twinkle. She showed me how to play it once and asked if i have any questions. I thought this is a bit weird, but again, i have not gotten any official lessons from her, so I don't know how she is yet. I don't want to make any judgement yet. Anyways, my question is, I'm doing the twinkle twinkle song (dude, i dont' know how much longer i can tolerate this theme), when I'm changing strings from A to E string, I can hear a the A slightly briefly before it hits the E, I guess this is a string crossing problem?? But the bigger problem is when I'm doing the third measure, DDC#C# BBA, when I'm switching from D to C#, and from C# to B, and B to A, it almost sound like an ornament sound or trill, I dont' know how to explain it, but I can hear the previous note briefly before i get to the next note. Is it because I'm not releasing my finger at the same time I finish the bow? I'm getting quite frustrated... any tips/help? Thanks!!

Re: teacher question, i guess it's still too early to judge which one mesh with me better since I haven't taken an official lesson with the 2nd one, but does she seem a bit too fast?

To me it sounds like you have 'self diagnosed' your problem, it does sound that maybe you have not released your finger in time with the bow change, do it really slowly and watch yourself when you do it, you will find the answer (same for string crossing problem).
As for the teachers, you are right in saying that maybe you should wait a little longer before you make your judgment. I personally don't like having tapes or stickers on my fingerboard and when my teacher got them out I told him to put them away and so he did. But this is a personal thing, you may not mind or actually like them. I like the sound of teacher number one better, mainly because she (or he? sorry) took more time with you and gave you instructions to practice bowing on open strings (an essential basic I think). But you see for yourself what you prefer.
well done so far!

I had this experience as well when I started out, I tended to slide off the string to the next one, changing from a A string to a C string did not sound good (Violin terms would be from E to G?)

Your probably not changing fingers in time with the bow.

Trust me, all this comes with time and practice. As for the teacher thing, my teacher is a combination of both. He didn't start me off with tape or anything (I already played serveral musical instruments at the time I started taking lessons) but I was bowing by the first or second lessons. Me and my teacher don't use the Suzuki method, we used a book made by Muller Rusch although I do practice on Suzuki book at home, and so far its been going ok.

When it is time to pick a teacher you will know which one you like most/more. The best teacher in life is yourself and nothing beats experience.

In one sentence: co-ordination and quickness of fingers.

It will take time to develop, be patient!