I would write these as follows:

We recommend that you book your flight early.

He recommends that you read the book before watching the movie.

I recommend that you go there.

I suggest that you go to the sport center.

See the definition recommend:

4. to advise; counsel; suggest ⇒ "to recommend that something be done"

I think it is usual to use that as a conjunction in these examples.
See the definition that:

7. a subordinating conjunction used to introduce:

So I would write them as follows:

We recommend that you book your flight early.

He recommends that you read the book before watching the movie.

I recommend that you go there.

I suggest that you go to the sport center.

Also check the definition recommend and you can see this form:

4. to advise; counsel; suggest ⇒ "to recommend that something be done"

I would write these as follows:

We recommend that you book your flight early.

He recommends reading the book before watching the movie.
(This is OK but you could also write "He recommends that you read the book before watching the movie.)

I recommend that you go there. (you going does not make sense.)

I suggest that you go to the sport center.

See the definition recommend:

4. to advise; counsel; suggest ⇒ "to recommend that something be done"

I would write these as follows:

We recommend that you book your flight early.

He recommends that you read the book before watching the movie.

I recommend that you go there.

I suggest that you go to the sport center.

See the definition recommend:

4. to advise; counsel; suggest ⇒ "to recommend that something be done"