As a muslim, is it a sin to keep my religion for myself, or do I necessarily have the duty to influence and convert as many infidels as possible during my entire life?
How do you know all your neighbors are infidels? People of the book and many many many others are not infidels.
But no you shouldn't just knock on their door and say let's read the Qur'an so you can convert sooner. You should be a nice Muslim and socialize with them following these instructions given by the Prophet:
- You must help him if he asks for your help
- Give him relief if he seeks your relief
- Lend him if he needs a loan Do not block his air by raising your building high without his permission
- Do not harass him
- Give him a share when you buy fruits; if you do not, bring what you buy quietly and let not your children take them out to excite the jealousy of his children.
- You must visit (and take care of) him when he is ill
- You must attend his funeral when he dies (and take part in burial arrangements)
- If he commits a sin, prevent it from being known
- Congratulate him if he is met with good fortune Grieve in sympathy if a calamity befalls him
Doing the above is among their rights. So you should do that first and foremost. If you socialize then I'm sure they'll figure out you're a Muslim. If they were interested and asked you questions about Islam then proceed further otherwise just keep socializing as you were.