Welcome! Please kick off your shoes and stay awhile!
As for anime recommendations I'll always point those who are still quite new to the anime world to the much older anime selection first.
A few to list are:
- Trigun (1998)
A series about a man called Vash The Stampede whom succumbs to amnesia before the series. During his travels to remember who he is exactly, he runs into two women who claim to work for an insurance company (they deal with compensating businesses, towns and people in general who were unfortunately at the wrong place, at the wrong time and suffered serious damages and/or injuries) and because Vash has a huge bounty on his head they decide to follow him (although he claims he has no idea who Vash the Stampede really is and makes no such claim that he is indeed the wanted man in this series) throughout his journey.
- Chobits (2002)
If you like slower paced anime and technology (robots, computers, etc.) I highly recommend Chobits. A series about a man down on his luck when it comes to money and relationships. While attending college and working his butt off to pay the rent and tuition he discovers what looks to be an obsolete persicom (a computer that resembles a human, kind of like a android so to speak) lying next to the trash outside. He drags her to his home and soon learns she is no ordinary Persicom but a Chobits.
- Akira (1988)
Now if you're into scifi big time, this is another anime you would enjoy. Akira is set in Tokyo, the year 2019. A small biker gang attending high school during the day and causing mischief at night. The leader of the gang, Kaneda and another member named Tetsuo are constantly feuding with one another only because Tetsuo feels like he has to be saved every time something goes wrong. He wants to prove to Kaneda that he doesn't need anyone in the group to help him. Later, Tetsuo comes to discover that he has psychic abilities but the downside is he cannot control them and the government soon learns of him. Because of what occurred during the past that nearly destroyed the city, the government takes Tetsuo against his will for experiments. Kaneda and the rest of the gang then try to go after him.
I have many more anime that I can suggest for you but this is a start. Some you can watch on FUNimation, Hulu and Netflix. For those that aren't on any of these platforms I'd suggest looking on YouTube or Google. Amazon may have copies you can purchase but be wary of fake discs.