Suggest somebody do something

Would it help if you put a "that" in there?

Mom suggested that ___ wear a jacket.

Mom suggested that I wear a jacket when going out to stop from catching cold.

Is the above sentence acceptable?

I'd say "to keep from" instead of "to stop from."

But I assume the rest of the sentence was not up for discussion?

(Yes, "I" is right.)

Can the sentence also be written as following?

Mom suggested to me that I take a jacket when I go out to prevent my catching a cold.

Wearing is not the same as taking, but both are fine.
To prevent my catching a cold is also fine (grammatically).

There is absolutely no connection between being cold or even being wet and cold and CATCHING a cold, of course. Total old wives tale.

Mom suggested ________ ( I, me) wear a jacket when going out to stop me catching cold.

i think "I" is fine,,perhaps better than "me"

mom suggested i wear a jacket when going out; to stop me from catching cold.

Hi hindss, and welcome to English Forums.

As we discussed above, "I" is correct (and "me" is not), but your suggestion of a semi-colon is wrong.

me

contiluocatching cold.

catching a cold.