"I think John will come to school today."
Placing 「今日」at the beginning of the sentence means that on this day you have the thought that John will come to school as opposed to having that thought yesterday.
It's a bit nuanced but it becomes clear once you use two temporal adverbs in the same sentence: Today I think that Jon will come to school tomorrow.
You can put any plain form of a verb or adjective before と思います.
For verbs, that includes the dictionary form (such as いく or みる), ない negative form, plain past tense affirmative that usually ends in -た, or plain negative past tense なかった.
ジョンは今日学校に来ると思います。= I think that John will come to school today.
ジョンは今日学校に来たと思います。= I think that John came to school today.
ジョンは今日学校に来ないと思います。= I think that John didn't come to school today.