The fastest way to get user's to your landing page is to pay for advertising, such as Adwords. But prior to doing so, I'd recommend that you install Google Analytics (GA) and track goals to see what % of the Adwords visitors are converting. Adwords may not be sustainable for you but it's a fast way to test the effectiveness of your landing page and messaging.
Paying for links is a very dangerous game. While the odds of your site being penalized are small this early, if it happens, it's pretty painful as you'll potentially be de-indexed altogether. If you are thinking of SEO as a strategy, start a blog and target specific keywords. Then have a clear call-to-action on your blog to your landing page. Obviously, your blog should be related to the product you are promoting. Use GA to see what keywords are driving traffic to your blog and try to rank higher for those with more targeted posts. note that this approach will take time and may not be right for you if you are just looking to validate your product - but it's likely a better long term strategy.
Another option is to find relevant discussion forums and contribute in a positive way. Many let you promote your site in some form (signatures, etc.) Just be sure to follow the rules of the forum.