The first companies to appear on the google search engine are the ones who have paid to be there. But after that it's a bit more complicated and you may need to employ a professional to oversee your on-line marketing strategy.
I think a lot depends on the name of your company/site and how reflective it is of the product/service you are offering. And if you are competing in a crowded market place with hundreds of companies offering the same thing that will have an impact too.
With regards to remembering the full address of a site, speaking from personal experience I rarely do; I either check it in the history, or do a google search making sure to use the actual company name (rather than the service it provides) to pinpoint it.
If they are customers who are returning to the site, most computers are set up to retain the history of sites visited so will quickly prompt the rest of the address.
But I take your point - the .weebly.com bit on the end is not an advantage. It's a free service and they cannot compete with a paid-for professionally designed site. Incidentally though you can register your own domain name separately and then just design the site through Weebly.
When weighing up all the options you have to think about how your customers will be driven to your site - will they be likely to already know about you (and therefore have your name), will they be looking for local generic suppliers (in which case they may go to Goldenpages online or google the area you are in) or will they search via service description, for example? These considerations should be taken into account when deciding on the website name.
If you think your product is going to be primarily sold to ad-hoc websurfers finding your site you should probably consider paying for professional advice.
If however, the website will be part of a multi-layered PR and publicity approach focussed on specific customers who can be targetted in a number of ways you might get away with something along the lines of Wordpress or Weebly.
Weebly has worked for me because I consider most people interested in the groups I'm promoting already have a high degree of motivation to find them. Either that or they have already heard about us and now just want to find out more information.
But obviously this is not the case for many people - especially when you are actually selling something.