You may want to try and use site templates. I've not tried to export a site template and upload it to SharePoint Online. It may not be accepted based on version differences - but since these are all declarative, there is a good chance it will work. Just remember site templates can't exceed 50MB, and they don't work for publishing sites.
Microsoft has migration API's and PowerShell cmdlets that are still in Beta, but avialable.
These however take a full export and will include list data, and requires some additional steps to prepare it for use.
You could probably cobble scripts together and do this your self too. You would need to download information regarding the site and lists and just use CSOM to build them new in Office 365. You can use PowerShell to do this, and there are a few modules you can find where people have already wrapped the CSOM calls into cmdlets you can use without having to know all the API calls or Rest stuff. http://blogs.technet.com/b/french_sharepoint_gbs_blog/archive/2014/04/08/how-to-manage-your-o365-sites-with-powershell-and-csom.aspx
If site templates don't work, I strongly encourage you to look at the tools @Waqas mentioned - they can SAVE you a ton of time and headache. I can absolutely recommend ShareGate as it is a dead simple tool to use, has PowerShell cmdlets, and their support is very responsive. MetaLogix is a good choice if you have complicated migration needs, it gives you a lot of flexibility in what you do during the migration. The support is not quite as good as ShareGate - but they have a huge KB library that addresses how to configure the tool for a variety of needs.