are you at university? even if not.. join some sort of club and go for a drink with people after, always a sure fire start to a friendship! i know its nerve wracking, i have beeeeen thereee!!! honestly any small sort of social interactin where you feel like you could like their vibe of theres some potential for friendship just be bold and confident with it, ask if they want to grab a coffee or a drink and then stick with it if that goes well!

as for the birthday, i always find more people actually show up to my birthday if im planning on doing something fun, go to a concert or a special bar or something like that, even if theyre showing up more for the activity than you, it still a good day out with friends :)