My SharePoint 2010 environment is holding Web-application and need to move for online SharePoint 2013.
And I could like to know is there any way to move the entire site content along with all the versions of document.
SharePoint Stack Exchange is a question and answer site for SharePoint enthusiasts. It only takes a minute to sign up.Sign up to join this community
One of the cheapest and reliable products is from ShareGate. It is not expensive and will do the required job.
They also have a trial version w/o limiting the functionality but in the documentation is mentioned that some files can be randomly skipped.
The only way is to build a tool of your own or buy/subscribe to a migration tool fit for the job. Several different tools have trial licenses you can try, but beware of that they randomly leave out documents and versions during trial license period.
Any customization needs to be re-built to fit SharePoint Online.
Google Search: SharePoint Server to Online Migration Tool
Their is no OOTB way to migrate from SharePoint on prem to SharePoint Online. I know we love on prem migration. Now you can move them manually or if you want o preserve all the metadata then you have to chose one of available 3rd party tools. I.e ShareGate, metalogix etc
Most of 3rd party tools offering the trail version, you can test and find a best suitable according to your requirement.