Best way to watch anime is Crunchyroll, it's legal and all in HD, but if you want to watch the newest series without Ads it will cost money. Crunchroll is available on Mobile, PC, Play station, X Box and a few smart TVs. If you don't want to spend money you can watch on bestanimes.tv or animerush.tv most of these sites don't have mobile support in older series, so you can watch only newer series.

Best animes to start getting into anime is the more popular ones then stem out to non-mainstream series, so usually Sword Art Online and Attack on Titan (Shingeki no Kyojin).