Dear chiquita79,
 
 ´First � is an Ordinal Number -  ´first �,  ´second �,  ´third �, etc.
 ´At first � =  ´At the beginning �;  ´In the initial stage, or in the initial stages �.
"At first, Nikki tried to be patient, but later ... ... "
 
 ´Firstly � is an Adverb.
 ´Firstly � = to give the first reason; the introductory point; the first point in an argument.
"Firstly, I did not take your ham sandwich; secondly, I don ´t like ham; thirdly, your sandwiches are disgusting; ... ..."
 
 ´In the end � =  ´eventually; or  ´on reflection �; or  ´after thinking about it �.
"I had a big argument with Petra but, in the end, I realised that she was right, so I apologised to her!"
 
1)  ´Finally � is at the opposite end to  ´Firstly � in an argument.
"Firstly, I did not take your ham sandwich; secondly, I don ´t like ham; thirdly, your sandwiches are disgusting; finally, I saw Chiquita eating your sandwich!"
 
2)  ´Finally � is used at the end of a long argument to make a final point, or at the end of a speech.
"Finally, thank you for inviting me here this evening.  Goodnight!"
 
3)  ´Finally � is used to end doubt or argument.
"I wish to state finally, that my Client is innocent of this crime".
 
4)  ´Finally � is used to mean  ´After a long time �, (especially when there have been problems or difficulties).
"Finally, we have paid all of our debts, and now our firm is in Credit again!"
 
I hope that this helps you.
 
Les