No such thing as best hosting company, only best for you.
Key criteria include stability, security, support (managed or unmanaged?), and of course, price.
You'll have to figure out how much resource you use at the moment, which should eventually help you select the right plan, hopefully from the right company.
Is there a particular reason why you must adopt HTML5?
friendly basically means easy to crawl by search engines, whilst keeping the ranking of search results of important key words in mind. For starters you can evaluate ease of crawl.