Are the English speakers place a great deal of importance on correct grammars?

For some people, proper grammar is important all the time. For others, I think most people, grammar is not important in casual, spoken conversation. In casual written conversation, such as texts or social media posts, it matters maybe a little bit more. In business settings, good grammar is preferable in spoken conversations and it is most important of all, essential really, in business written communication.

Go - is happening right now, in the present

Went- already happened, it's in the past.

"I go" just isn't proper. It would be "I am going to go or I will go" because "go" isn't current. When you say "I went" that means that you visited the park in the past. If you said "I am at" then that means you are currently at the park.

It depends *which* English speakers. Of course “I go to the park last Sunday ” is wrong, and every native English speaker will realise it. But people you are just chatting to probably will not think it very important, whereas someone interviewing you for an office job probably will.

What we place an importance on is communicating clearly, so grammatical errors are an obstacle to that end.