"I want to drink too."
Translation:Я теж хочу пити.
That's quite a complicated topic, as there many types of adverbs and we have to speak about them separately
The most common place to put adverbs you mentioned is before the predicate.
Я теж хочу пити, Я дійсно хочу їсти (I really want to eat)
Теж can also be put at the end of a sentence, but definitely not at the beginning
Adverbs of time can also be placed in the beginning or in the end of sentence without change in meaning (emphasis may slightly change):
- Зараз я буду снідати
- Я зараз буду снідати
- Я буду снідати зараз
(I am going to have breakfast now)