Depending on the complexity and competitiveness of the terms you're hoping to appear for in search results, it can take months and lots of hard work to show up.
The first step to helping your site show up in search results is by creating a Google Webmaster account and submitting your sitemap. The good news is that Weebly automatically generates sitesmaps for you that can be found at yourwebsite.com/sitemap.xml
The second step is by making sure your website is mobile friendly. The even better news is that the latest templates from Weebly are built to be responsive, so your website should be mobile friendly at publishing.
The third step is by creating regular, relevant, and interesting content on a blog. Write content that will answer questions for your prospective customers about things they might be interested in (e.g. a diaper blog might write on parenting tips). Google has made it repeatedly clear that a blog is the best way to get ranked/indexed.
I hope this helps, good luck with your website!
