Power-Level! (Raise your level by 50.)
Remember that we get daily rebirths for now so you can possibly do this each day you get it. Various Taillteann shadow missions are quick to run for the experience, like Battle for Taillteann II, or may be safe options like Dorren's Request. There's also ancient hunting in Scathatch or Iria.
Dreaming of Weaving Mastery (Use the Weaving skill 30 times.)
Unless you have a sheep and various gathering upgrades, it's quickest to pick up 150 cobwebs from spiders in Tir's graveyard. Then go to the spinning wheel outside the Tir general store and try to make 30 thin thread balls. It needs attempts, not successes, so low a success rate will allow you to do it with less materials (as you can get a refund on failure but not on success).
Never Do Tomorrow What You Can Do Today (Clear today's mission once.)
I assume this means to complete one of the daily shadow missions. There's one for Tara and one for Taillteann each day, and they'll be listed first on the board.
Dreaming of Metallurgy Mastery (Use the Metallurgy skill 30 times.)
You only need attempts, so you can stand on the same spot and try over and over, even if you're not standing on sparkles.
A Solid Investment (Upgrade An Item 5 Times)
Grab a Dagger and get 25 proficiency on it. You can do 5 upgrades of 5 prof each costing 500 gold each at Edern. Anybody know something that needs less prof?
Well-Armed Wanderer (Upgrade a weapon 3 times.)
Either bugged, or needs special upgrade attempts.
Proficient Field Boss Hunter (Defeat 3 field bosses.)
Try for the Dugald/Dunbarton ones that involve multiple entities like the Black Raccoon Family if you're not OP. You can kill random monsters in the area until you get a Fomor Command Scroll, use it to see when the next field boss will spawn. They'll spawn anyways, the scroll just gives everybody a heads-up at the top of the screen. This might invite competition though.Black Leopards in Raspa count.