Since you are ***migrating*** to new server, you don't need to bring SQL Server 2008 into the mix.  

As stated by SQL Server expert [Paul Randal](https://dba.stackexchange.com/users/94938/paul-s-randal), [You can upgrade from any version 2005+ to any other version](https://www.sqlskills.com/blogs/paul/you-can-upgrade-from-any-version-2005-to-any-other-version/).  Paul states:

> There’s a persistent myth that you cannot upgrade a database to a
> version more that is more then two versions newer.
> 
> It’s not true.
> 
> You can backup+restore (safest way) or detach+attach (not a safe way)
> from SQL Server 2005 (or higher) to any other newer version. **You
> just can’t do an in-place upgrade to the latest version from SQL
> Server 2005.**

As for changing the comparability level, I'd recommend not changing that for several weeks and let the dust settle from the upgrade.  After you're confident that everything is functioning properly from the upgrade, why not bump the compatibility level up to the maximum and monitor performance.  You _might_ even see an improvement in performance.  However, compatibility levels above 120 (SQL Server 2014) bring the new cardinality estimator into play, which might present some performance problems.  If you want, you can be more selective in your testing of the new cardinality estimator by using `OPTION (QUERYTRACEON 2312)` on selected queries.  You can always reduce the level should problems arise.  Do some internet searches on _sql server 2014 cardinality estimator issues_ to familiarize yourself with possible issues.

At some point in the future when you upgrade to a later release of SQL Server, you WILL have to eventually increase the compatibility level.

Finally, check out [Upgrade SQL Server](https://learn.microsoft.com/en-us/sql/database-engine/install-windows/upgrade-sql-server?view=sql-server-2017) for a checklist of items to consider for upgrading SQL Server.