Similar meaning, but less negative:
* Brag
* Show off
* Vaunt
* Extol (emphasizes praising)
* Tout (emphasizes promoting something)
* Celebrate (emphasizes positive accomplishment)
* Flaunt (emphasizes showing off possessions)
More negative connotations:
* Blow one's own horn
* Trumpet
* Swagger
* Strut
* Grandstand
* Pomp (more formal)
* Braggadocio (formal, refers to excessive boasting)
Neutral words for mentioning achievements:
* Mention
* Describe
* Relate
* Explain
* Highlight
The best word choice will depend on the specific context and the tone you want to create.