You can view an item's HQ stats by pressing a button to toggle what the tooltip shows, I believe it's R2 on the controller. No idea what it is on keyboard.

About gear rarity, it actually means VERY little. This is not Diablo or Borderlands, in this game it is the level of your item that matters. Specifically item level, which can differ from equip level.

White gear is your commons. The vast majority is craftable, but there are some pieces here and there that cannot. It should be noted that for the craftable gear, high-quality gear is the baseline that all other gear is compared to. Normal quality takes a step back. For levels 1-49 equip level will always match item level (with the exception of "damaged" gear from dungeons, see below), but past level 50 the item level jumps up due to the rise gear took when level 50 was endgame.

Pink gear is aetherial gear. This is the only gear with randomness, and drops from early dungeons. It will share a name with a piece of crafted gear and will have the same PRIMARY stats, but instead of the secondary stats the base piece has, it will be a random assortment of them. The item level is the same as the base item as well, but the equip level matches the level to enter the dungeon, which may be a few levels lower. There's also level 60 pink gear that comes from Diadem. This is mostly the same as aetherial gear, but it has an equip level of 60, there are different tiers of item level depending on the chest it came from, and the appearance is also randomized.

Green gear is your standard "rare" gear. For levels 1-49, green gear has a few item levels more than its equip level, while for 51-59 it is exclusively dungeon drops and grants gear for odd levels (while crafted gear handles evens), with a fairly linear progression as equip level rises, regardless of white or green. Green gear drops from dungeons, or is crafted using materials dropped from dungeons (the "damaged" white gear from dungeons typically), while green gear at levels 50 and 60 may be crafted using materials that can only be gathered at certain times of the Eorzean day, and represent the various tiers of crafted gear once one has hit endgame (The reason a lot exist at 50 was because that was the cap for a while). There's also a few green items that can be purchased with special currencies, such as from grand companies or PVP.

Blue gear is... precisely the same as green gear except for how it's obtained. Blues are more "significant" than greens, and are either purchased with tokens at endgame, or dropped by raids or primals, orobtained via quests and achievements. Typically, blue gear represents more of an accomplishment than green gear (which can usually either be bought or obtained from an easy dungeon), and also has a fair amount of lore attached most of the time. Still, the difference only matters for showing off, again, item level is all that matters here. Oh, there ARE some blue items that drop from dungeons, but those were formerly obtained with tomestones that have since been outdated.

Purple gear is used EXCLUSIVELY for relic weapons. These weapons are MASSIVE grinds, but tend to be quite flashy, have a TON of attached lore, and even allow you to allocate the secondary stats as you wish past a certain stage. They're also a way for players with more time than skill to have a decent item level weapon... or to show your dedication.