Litespeed works instead of Apache, but switch is instant without any changes, you can find litespeed hosts at there site Litespeed partners You will be able to find plenty of shared and VPS/Dedicated hosts listed there who are offering Litespeed, some have litespeed as default (Like Crocweb, Stablehost) on some this is an option, as unlike Apache which comes free, litespeed is paid server so hosting companies keep it as an option to the customer.
I have heard good recommendations about CrocWeb (They have litespeed and cloudlinux), but because they offer just personal accounts, so you may would be better off with Stable hosts' enterprise class hosting starting from $19.95, they are on RAID-10 SAS discs, so no performance bottlenecks coupled with litespeed servers. it will be powerful.
you dont have to worry about the servers running at backend, as you'll be given CPanel to work with to use Moodle.
I have had used bluehost/hostmonster (same company) shared personal hosting with Moodle 1.9 in past, and it worked well. if you want to try them out, then start with there Shared PRO accounts, starting at $24 a month, and with bluehost you have ease of switching/upgrading at time as when you feel like needing it, all you have to do is tell the staff what time you want the switch, and it will be over even before you have managed to make yourself a cup of tea.