Welcome to the forums!
These are the anime I'd recommend to a newcomer in order of different genre:
Mech/Space Mech- Gurren Lagann
Sci-fi- Ghost in the Shell: Stand Alone Complex & 2nd Gig or Psycho-Pass
Slice of life- K-On! or Hidamari Sketch
Fantasy- Little Witch Academia
Comedy- Space Dandy
Drama- Anohana!
Action- Black Lagoon, Second Barrage & Roberta's Blood Trail
Sports/Hobbies- Yuri!!! On Ice or Akagi/Kaiji
Game/MMO- No Game No Life
Romance- Eureka Seven (this is technically sci-fi and mech but a lot of the focus is put on the developing relationship between the two main characters)
90's- Berserk
Post-Millennium- Naruto & Naruto Shippuden or Fullmetal Alchemist Brotherhood
Ditto
Seriously Billyuk you should visit the Anime for Veterans and Beginners thread for some off the cuff recommendations.
Anime for Veterans and Beginners
Also think of looking to Crunchy Roll for some free anime. It is a very ready source for those looking to bugdet and (re)find their feet in anime.
One thing I always suggest to new fans, something Captaaainuniverse and yourself have already hit on. Slow your roll. Too much, too soon will lead to a burn out (even if you are enjoying yourself) and being the old school sort that I am, it is a hobby that can get expensive (330+ titles, over 1000 dvd's and something short of a university tuition spent on such a collection. It was a hell of an education).
Find your feet, be willing to try new things, but watch what you buy, literally and figuratively. The backlog thread should give some idea of how insane things can get and hopefully keep things a bit more rational for you. If you need help give any one of us a shout and we'll certainly try.
I will throw this up as a PSA to Newcomers. Avoid Thai or Malaysian Specials on Ebay or Amazon. They are cheap for a reason. Even if overall quality isn't high priority, there is a good chance you are shunning the hand that feeds you, not to mention that odds are the titles you're buying are already available for free or on subscription on Netflix or Crunchyroll. Pay the producers their worth.
Finally I'll throw this out to you for some ideas (bear in mind I focus on older anime although there will be some newer ones)
Mech/Space Mech- Bubblegum Crisis OVA/Macross Plus/Battle Angel Alita
Sci-fi- Gungrave
Slice of life- Summer Wars
Fantasy- The Girl Who Leapt Through Time
Comedy- You're Under Arrest (if you're feeling bold try Excel Saga)
Drama- Cowboy Bebop (This title fits nearly all these Genre as well)
Action- Redline/ Riding Bean/ Gunsmith Cats (inc the manga)
Sports/Hobbies- Bamboo Blade
Game/MMO- .hack// series
Romance- Oh! My Goddess.
80's- Ninja Scroll/ Castle of Cagliostro
90's- Berserk/ Slayers/ Cowboy Bebop (really can't emphasis this title enough)
Post-Millennium-Soul Eater/Fullmetal Alchemist Brotherhood