We offer a suggestion of an action.
I suggest that you eat lunch now, because the cafe will be very busy later.
To construct a suggestion first create an instruction:
You eat lunch now
That's grammatical but sounds rather like an order. In many friendly circumstances we'd soften that:
You could eat lunch now
This has now become more of a suggestion, we've made it clear that the other person is free to make their own decision. We can take that a step further and explicitly state that we are making a suggestion:
I suggest that you eat lunch now
notice here that we have used that. Applying this to one of your examples
I suggest that you go there tomorrow
The big difference from your formulation is that we do not say
I suggest you to
We do not use you to. The person is not the object of suggest, we are not suggesting a person, we are suggesting an action.
It is common, especially when speaking to miss out the that so we get
I suggest you eat lunch now
I suggest you go there tomorrow
It may help to recognise the structure if you pretend that the that is actually there.