What you should do in terms of SEO for a brand new website.
We all have websites or we are in the process of setting one up. I know this because why else would you be here reading my post lol
When setting up a website you'll want to definitely focus on your on page optimization as your getting each page up and running. If you're using a wordpress theme then you can go through all your pages one by one using the Yoast plugin to make it a bit easier on your brain. On page is definitely important, but just having on page optimization won't get your ranked for those difficult keywords unless your website goes super viral. And getting a website to go super viral is almost impossible if you think about it.
Your off page optimization is the peanut butter to your onpage jelly optimization. You can't have one without the other, it just doesn't work well. I do love my peanut butter sandwiches, but I'm not here to talk about food ;)
When you get started with your off page optimization for your new website you'll need to be careful and take it slow. You're going to be posting links, or paying someone to do it, so you can increase in the the SERPs (search engine results pages) for specific keywords in order to pull in loads of traffic. But if you go all out guns blazing right after you get finished with your on page optimization, you're not going to last long.
Start with some forum posting and have your links in the signature so that everyone can see it, even the search engines. This will get you on the radar since a lot of forums get spidered on a regular basis. You can also get active on facebook and LinkedIn to get indexed almost overnight.
After about a week of this I would suggest posting to question and answer websites to get some links. Now you won't be able to post your links right away since you'll need to build up your profile. So go around and answer questions until you level up your profile to a point where you're allowed to post hyperlinks to outside sites. For this I usually pay someone here on SeoClerks to do it for me I tend to pay Cristian for this type of work since he does a great job at it
After another week of this I would say to pyramid, or tier, these links in order to push even more juice through them. To do this you can buy a GSA type service on SeoClerks and get it going fairly quick. You can compile all of your posts and have the GSA campaign blast links to them.
Keep posting manually to your forums and keep up with the Q&A type sites because you'll want to build an authority over time. After a couple months you can branch out and start doing much more linking and get a bit crazy. But until then you should be playing it safe and don't push it too much.