The word comigo can be seen as the contraction of "com" and "mim, so You use it for constructions with "with me".
An example would be "Are you coming with me?", which translates to "Vens comigo?", or alternatively "Você vem comigo?"
The other contractions are, following the typical person order
Comigo (eu)
Contigo (tu)
Com ele/ela/eles/elas
Connosco (nós)
Convosco/com vocês (vós/vocês)
This however doesn't work for the reflexive constructions "with myself" and so fort, which is slightly different. Although this construction is weird in English, it's possible in Portuguese such as is the case with "I can only count on myself", which in Portuguese would be translated as "Só posso contar comigo mesmo"
Happy learning!