No, you can't. And indeed - 1 or 2 star artifacts are almost worthless, even for new players, you'll get 3 and 4* artifacts soon enough. Maybe you use 2* gear for a week or so, but probably not longer - and don't bother investing gold into leveling it beyond level 8 or so. 

An artifact's rank (number of stars) and rarity (color) is locked in.  You can "upgrade" its stats, though.

Rank and rarity are less important than stats.

As an extremely quick-and-dirty guide, you should look for:

-Gloves with Crit Rate or Crit Damage as the main stat.

-Chest pieces with Def% or HP% as the main stat.

-Boots with Speed as the main stat.

-Weapons, helmets, and shields all have the same main stats (flat ATK, HP, and DEF, respectively), so on those ones you have to look for substats.  Generally you'll want synergy, so look for weapons with ATK%, helmets with HP%, and Shields with DEF%.  Speed is always good, and it's unquestionably the most important stat in the PVP arena unless and until you ascend to the absolute highest ranks of this competition.  And you also need to look for ACC as a substat--most champions require ACC to land their debuffs, and will be just about worthless without it.  Crit Rate and Damage are good substats, too, but of somewhat lesser importance.  "Flat" (i.e. non-percentage) ATK, DEF, and HP aren't usually all that helpful, and generally you'll want to junk any piece you find that has a lot of them.

-The foregoing are just rules of thumb for recognizing good equipment, but you might need to look for other things to build specific champions (e.g. it's often worth spamming as much HP as possible onto HP-based champions, and champs that don't need ACC give you a lot more freedom to maximize other stats).

-Remember that ATK and ATK% don't do anything on a champion whose attacks do damage based on some other stat.  If you see that a champion's damage is DEF-based, don't bother putting any ATK on that champion.

When you're ready to "upgrade" stuff... Well, come back, as there's more to know.

Oh, I agree with the post above that you shouldn't try to upgrade 1 and 2 star artifacts.  You might find some decent 3 star pieces in the very early going, but with rare exceptions (e.g. the free shield gear you get in early daily login rewards), it's not worth upgrading these past level 8, or maybe level 12 if you get lucky upgrade rolls.

For the most part, you won't want to upgrade 4-star gear past level 12 either.  If you're going to take something all the way to level 16, you have to be sure it's worth the investment, as on average it'll take at least a million silver to get it up that high.