"tsuite". At tsui-, do you combine it and say it like tsoowee, kind of like your saying "Sweet"? or would you just say tsu then just say eete right after? You're welcome to ask me to clarify if I'm not making enough sense and I'll try to explain it better.

There are no blended vowels in Japanese (these are call dipthongs, btw, when your slide from one vowel to the other.) Whenever you have a case like that it's one vowel and then the other.
It takes some effort both to do it and to hear it for beginners coming from and English-speaking background where we blend our vowels all the time.

To play it safe in spoken Japanese, I would pronounce the word in full at a normal rate of speed when talking.

So something like T'soo-ee-teh (but at a faster pace than slowly sounding each syllable out) with a hint of the T sound being pronounced with the tsoo, but with the つ and the い being pronounced in a fluid motion similar to the word "Sui (water)", just with the added slight T sound proceeding it.