SEO or search engine optimization is a process that helps rank your website at top place in google or other search engine results. It involves on page and off optimization that brings organic traffic on your website..
SEO is a good medium but i will suggest you to try social media for this process. Facebook, Twitter, Reddit etc are good for these kind of websites. If you can spend some money on social media, then it will be more effective for your website.
Both Twitter and facebook are good medium for promotion but my vote goes to Facebook. It is more effective in generating leads. But firstly you need to have large number of followers in both social media giants then only you can succeed in successful promotions.
I prefer doing it alone. If required i hire a freelancer developer to build something special for my forum but otherwise i just invite people to my for discussion. And yes i agree with you using facebook ads, google ads or twitter goods are good for this promotional purpose.
Yes, Social Media helps in increasing reach of your forum. But you need to make a good strategy and follow it to make people aware of your forum. One more thing, hashtag are important while using social media, using them wisely can pay a lot more dividends.
low quality content, Not Policing your forum for Bad Actors, Not using appropriate keywords and meta tags - These are things that can really hurt SEO for forum. If you have any more suggestion, then please reply.
Giveaways are okay as a temporary measure but keep your focus with Social Media Marketing. Use Facebook, Reddit, etc to promote it. Also you can use services of good SEO as suggested by one of the members above.