I had a boy two years ago, and he will be goingo to nursery school next year.

While your original sentence is fine, I find that "will be going" is more natural. Between "will go" and "will be going", there seems to be a very subtle difference in tone; the former might come across as a little more forceful or assertive, while the latter would sound more neutral. Generally speaking, "will be going" is likely the more common way to say it.

The tuition fees of public nursery schools are lower, and the quality of the teaching staff is more guaranteed.

Just looking at this sentence, it's not clear what exactly it is that's "guaranteed", so you should state it explicitly. Judging by the following sentences below, I'm guessing that you were referring to the quality of the staff. I do get the sense that you're trying to convey the message that the teaching staff in the public schools are "safer" and more "reliable".

However, there are more children in each class, and the teachers' careattention is limited.

In my experience, "care" and "limited" don't typically go together in this context. "Attention" might be more appropriate.

On the other hand, private nursery schools are more expensive, and there may be some unqualified teachers in them, making it hard to distinguish the good teachers from the bad.

(1) Alternatively: "and they may have some unqualified teachers, making it..."
(2) It is unclear what it is exactly that's hard to distinguish. Similarly to your prior use of "guaranteed", this should be made known to the reader explicitly. I took a guess at what you meant to say, but perhaps some native text can provide further clarification.

My favorite is a small but warm private nursery school, but the fees and the distance are issues.

"Favorite" is fine if the school you're referring to is one that you know exists. However, if you're describing what you IMAGINE to be the best school, you'd say: "My ideal (school) is a small but warm private nursery school".

I had a boy two years ago, and he will start goingo to nursery school next year.

I think this is more natural

The tuition fees of public nursery schools are lower, and the teaching staff is more guaranteed.

It's a bit unclear what "The teaching staff is more guaranteed" means. You should mention what is "guaranteed". Do you mean that a low student to teacher ratio is guaranteed? Or that the quality of teaching staff is guaranteed?

However, there are more children in each class, and the teachers' care is limited.

You could also say "the teachers have limited time with each student".

On the other hand, private nursery schools are more expensive, and there may be some unqualified teachers in them, making it hard to distinguish.

Similarly, it's unclear what the teachers/private nursery schools are "hard to distinguish" from. Do you mean it is hard to distinguish if the teaching is good or not?