Years ago, I dated a guy via a friend of a friend at an anime convention that like me, very obviously loved anime. We eventually parted ways romantically but still stayed friends and still hung out together at conventions. At one con recently, he started talking about his new girlfriend who was really into Western geek stuff but hadn't checked out anime yet. Now, I (and many people my age) were introduced to anime from the Toonami days - Sailor Moon and DBZ, then Outlaw Star and Tenchi Muyo!, with Adult Swim coming along with even more anime that really made me realize that animation was a media for storytelling, not a "kid's only" exclusive means of entertainment.
But at this moment I realized - not everyone grew up with those shows when they were 13! It's so natural that I grew up loving anime because I feel like those shows were accessible to me at just the right time/age. This got my ex and I talking about what shows to introduce to her and really for anyone that is wanting to watch anime for the first time.
There's a LOT of different styles and types of anime - you have the "classics everyone should watch" and "the unknown gems that never got subbed". For the sake of keeping one's interest, I haven't included any epic saga that has spanned almost a decade and episode counts in the hundreds or is 90% filler/"Monster of the Week" episodes (sorry, InuYasha!) The following list might not have every staple in anime history and certainly isn't a "Best Anime of All Time" list, but are some shows I recommend watching just to get your (or your friend's!) feet in the water.
1. Cowboy Bepop