It is possible to get good first page rankings for a new website in 2 or 3 months IF you don't have a lot of competition locally and IF you do a good bit of social media work, some blog writing, and make sure your new site is listed in a few online directories. I build websites for a living & manage it fairly regularly - - but I also see ppl who don't make the effort with blogging & social media who are still underwhelmed after twice that time. You get out of it what you put in!
Remember that "content is king" - Google loves to see fresh, well written, unique content - and regularly updated content too. So a website with more pages & more words on each page will stand a better chance of getting good organic rankings than a smaller minimalist site (That's a bit over simplified, but the principle holds) and adding new content every week with a blog post can work wonders too.
The more competition you have & the less work you do, the longer it will take, and the lower your final ranking will be - - but I would nearly always recommend putting the effort (and even money if required) into content creation rather than paying for Adwords. In my experience I've always found off-line advertising to give better return on investment than Adwords.
The one area where Adwords can't be beaten is if you need online exposure for an area where search traffic is low. There's no point having first page rankings if your search term only gets 3 searches a month,.. and Organic rankings are passive, relying on search volume.
Adwords is an active system - it places your listing in related search results, going out & getting traffic from searches that you would never have appeared in organically.... but because some of these "related" searches are actually quite different from your targeted search phrase, the click through rate and bounce rate are rarely as good as organic results...
