Hi, Ella.  I am pleased that you have taken the time and shown interest in asking me to confirm your understanding of the lesson.  When responding, I usually put my comments in brackets [   ]. to make it easy for you to identify them.  OK, here we go...

Hello Lucky! I'm ella :)
I wanted to review the previous lesson.
I summarized the lesson of suggestion myself and i would greatly appreciate you check and correct ! [appreciate IF you WOULD check IT and MAKE CORRECTIONS]

1. Suggest
There are 2 ways to use verb "suggest"
1) suggest to me/you ~
- I cannot say "he suggested me that", should say, " he suggested to me that i should~"
in this case, "to me" is indirect object, "that~" is direct object. [your understanding of this use of 'suggest' is CORRECT]

2) suggest me (for the job)
this is the sentence that no need for "to",
ex : later i realized that he suggested me for the RA job.
in this case, i am not the direct listener of suggestion.  [also CORRECT. 

Note that there is an alternative to the word 'suggest'; you might instead say 'he recommended me for the job'.  Recommend can be used in the same way as suggest; 'he recommended to me that I join the club', or 'If you are looking for a good hobby, I recommend that you join a club.'  'Plese recommend a doctor for me.', and 'I recommend that you see Dr. Smith.']
thank you  [you are so welcome, Boss.  I love to anwer questions!  See you on Thursday!]