We all have to start somewhere and it’s not easy with so much great anime out there , it makes it a little hard to choose what we want to start with.
There are many genre’s to choose from and depending on where people start they may have a fractured view on the world of anime , especially if the first is not to their liking , so the first is quite important.

In that regard it is important that the first should have a large enough range so that you can find some part of it to your liking and from there find out more about your own taste in anime , knowing what you like in movies series and books helps quite a lot as well as you can direct yourself more towards what you know you like and develop your palate from there.
For instance , if you like romance you start off watching romance and as you go on you diverge into romance + mech , romance + supernatural , romance + SoL and so on so that you can find something that you enjoy and from there explore the new genre that you have an interest in and grow more interest in anime as a whole.

No one ever stops changing so your current favorite series or genre might not always be the same but if you explore and find more that you enjoy you will never find yourself unable to enjoy anime as there will always be something that will interest you.
Here are a few anime I believe are great for new anime fans or people who want to start watching anime.
[In no particular order]

Shigatsu wa Kimi no Uso

Drama , Romance , Music , School
This series shows a lot of emotion and is a great starting point if you enjoy drama , music or romance. The music is also brilliant and if you enjoy the music genre it will not disappoint.

Magi -The Labyrinth of Magic

Action , Adventure , Fantasy , Shounen
This series has a lot of action , adventure and magic and is a great starting point if you enjoy fantasy. In addition to it’s brilliantly done fight scenes it has some powerful scenes that tug at your emotions.

Mahou Shoujo Madoka Magica

Drama , Magic , Psychological , Thriller , Magic Girl
This series is a great place to start for people who want to get into the Magical Girl genre as it is both a brilliant Magical Girl anime and has a serious tone to it that allows even those that are not fans of the genre to truly enjoy it. It is also a brilliant thriller and psychological theme anime and should not be overlooked even if you doubt Magical Girl anime.

Bakemonogatari

Mystery , Romance , Supernatural , Vampire
This series is a grate starting place if you enjoy mystery or the supernatural. The use of the vampire genre is not over used , though not forgotten or neglected and the series is that much stronger because of it.

Love Hina

Comedy , Ecchi , Harem , Romance , Shounen , Slice of Life
This series is a great place to start if you enjoy the comedy , SoL , ecchi , romance , or harem genre’s. The comedy is great and the harem and ecchi scenes are done well all while being grounded in real life so that it is not too outlandish.