How to move a 80GB MySQL Database with least effort and least time offline?
The main problem with MyISAM is that you cannot create a copy of the database without locking the tables (otherwise the copy may be inconsistent). I would recommend you to research a potential migration to InnoDB.

You will have to execute FLUSH TABLES WITH READ LOCK in order to lock all tables in read only mode, then copy all the .MYD, .MYI and .frm files from the filesystem (basically, your entire datadir). Once you copy those files, you will want to get your binary log coordinates so you can setup replication with SHOW MASTER STATUS. If you cannot setup replication, you will have to leave the server in  read_only mode while you setup the other one. Otherwise, UNLOCK TABLES;. In previous versions of MySQL, you had the mysqlhotcopy script that automatized some of those tasks for MyISAM.

EDIT: As you have some InnoDB tables, you must be sure to copy the whole data directory (including ib_logfile* and ib_data*ones), and not only single files- otherwise the copy won't succeed.

Setup the new server copying the files to the right place, start the new server, and you can now setup a replication between the old server and the new one. Once you double check that the setup is correct and the key_buffer/buffer pool is warm, you will put the old server in read_only mode and failover your applications to the new one.

The idea is to avoid mysqldump, which is slower for exporting, but above all, for importing, although it is easier to use. That it is what it is internally executed with phpMyAdmin's "export/import database" feature.

I know this is a high level overview, but I recommend you to start reasearching the links I mentioned and about MySQL replication and asking additional specific questions if needed.