Ah, a new soul to corrupt. Good, good ...
I agree with the general census of watching whatever tickles your fancy at the moment - being new to the scene probably means being less picky about specific elements of a show, thus it's probably harder to "miss" with a choice - everything seems new and fresh after all. I'd probably advise against the mainstream shounen titles like Bleach though ... while having a reasonably good start, a few hundred hours later I could only wish for all the wasted time back. Ah well, one way how to learn not to play with fire is getting burned
Generic musings aside though, let's conjure up some actual suggestions along the fantasy/romance/action lines. I'll try tossing in a few older shows (as in, not from the most recent two-three years) that otherwise might take a while to stumble upon. (I'll keep the summaries brief, as longer ones can be found on the other end of the links
)
Shakugan no Shana - a fairly popular title, third (and last) season is currently airing. The main character finds himself assaulted by a monster and "killed", only to find out some disturbing truths about himself and the world as a whole. Obviously fantasy and action, relationships play a fairly significant part as well.
Elemental Gelade - mostly adventure/fantasy, though with a surprisingly well developed and conclusive (<Important! A
conclusive romance is one of the things you will learn to appreciate as you watch more and more anime
) romance between the two leads. Generally more light-hearted than most titles on your list, though well worth a look.
Chrono Crusade - a story about an unlikely pair consisting of a Sister and her demon partner, on their mission to hunt stray supernatural oddities that develops in scale as the show goes on, with a very heartfelt conclusion. Fantasy and action is a given, as well as the developing feelings between the two leads.
Utawarerumono (Yes, that can be a mouthful!) - A man finds himself waking up in a village with no memories of who he is, and a half-mask stuck to his face that wouldn't come off. Despite lacking memories he finds himself talented in many things that help out said village, eventually involving himself in military struggle as well. There is no lack of fantasy and action in the setting, and something on the romance side to chew on as well.
Scrapped Princess (Among my favorite animes overall to this day) - a story about princess who was prophesized to bring about the ruin of humanity on her 16th birthday, and thus to be disposed of as baby. She survived the attempt though, and the story follows her and her protectors as they flee the ever increasing pursuit as her 16th birthday approaches, all the while the truth behind her curse slowly begins to unravel. Fantasy and adventure indeed, kinda light on the romance though.
Shounen Onmyouji - If you can survive the lack of focus on action, I found this to be a quite good and entertaining story about exorcism set in ancient japan, focusing around one of the emperor's best exorcists, or more precisely the grandson of said exorcist, obviously spirits of all kind, and the court princess he eventually develops a relationship with.
Blood+ - reasonably lengthy show about a battle against form-shifting humanoid creatures in the shadows of society. Get your dose of secret agencies, vampire-esque (or not) monsters and sword-fighting (not only) here, wrapped up in a fairly grim atmosphere.
I guess I'll stop here before the list grows too long, but not before throwing in a suggestion of something slightly different -
Shinigami no Ballad. Features 6 short stories featuring a death-god called Momo, on 6 separate ventures that (obviously) deal with death and the impact Momo leaves on those she appears to. Some of the stories feature romance as theme as well. A quite melancholic title overall - certainly not something to sate one's action needs with (as there is none), but quite interesting all the same - can't hurt to have something different from the mold, to feel out what other themes you might appreciate in shows.
On another note, I'll second
Higurashi - a fantastic mystery/horror show, one which will keep you guessing and intrigued from start to end while you try to piece together the pieces and figure out what happened, why it happened, and what triggered things happening differently this time around. Don't let the cutesy art throw you off, this title is well worth the time spent watching it.
As for other things on your list ... Since you watched Clannad and plan to watch Air, I would suggest adding
Kanon 2006 to said list - all three written by the same person, produced by the same anime studio.
Tsubasa Chronicles is a fairly good show, but is sadly incomplete. Chances are you will enjoy the various episodic adventures (most spanning 2-3 episodes) and various worlds the cast visit during their journey, and the music is simply great, but the lack of a conclusion might be a negative. Still worth giving it a try though.
Mushishi is an impressive piece of work, but it's unlike most other things on said list. Don't expect any action or fantasy (in the flashy sense of the word) here. It's about a traveling Mushishi (a person who can see the supernatural, often bug-like organisms (Mushi = bug) who interact with reality behind the scenes and often leave unexplainable effects in their passing), Ginko, and the various mushi related cases he encounters during his travels, and helps solving with his expertise in these phenomenon.
Do give it a try, but you will have to appreciate the silent beauty and slow pace of the show to derive enjoyment from it - the issues here get solved by analyzing the situations and applying some form of intellectual/spiritual/medical countermeasure to them, not by swinging swords and tossing fireballs around. A very worthwhile show non the less.