go straight (somewhere)

"Go straight ___" is usually used to talk about going to a place without stopping or going somewhere else first:

I went straight home after work.

But it's also used in the phrase "go straight to sleep", which means to go to sleep immediately after something. Parents use this phrase when they're punishing their young children:

If you don't behave yourself, you're going straight to sleep after dinner. Do you understand me?

go to sleep

When you're talking about sleeping, there are several stages:

• When you're awake and then start sleeping, it's called "going to sleep".
• After you have gone to sleep, we say that you "are asleep".
• Then you "wake up".
• Once you wake up, you "are awake".

There are other actions that have similar stages. For example, you "put on" clothes, then "wear" them, then "take them off".

(do something) as soon as (something happens)

The phrase "as soon as" expresses something that happens immediately after something else:

He called as soon as he heard the news.

I went straight to sleep as soon as I got home.

Another way of expressing a similar idea is "when":

I'll start dinner when I finish writing this e-mail.

But you use "as soon as" to emphasize that you're doing it as quickly as you can. So this phrase is useful in situations where you're making an excuse for being late.