This is my first time ever playing DAoC. I'm a veteran to MMORPGs in general, playing since the open beta of ragnarok online (early 2000s) but I have to say... this game is absolutely confusing to me. Either I've been spoiled by modern UIs or I the new games dumbed me down.

I started as a hibernian, and I don't even know how to self-target to use my healing skill, how to kill those "frogs" around me, how to see if I have available quests... I believe I delivered an item to an NPC after crossing a bridge (took me 15min to figure out I had to drag-drop the item onto the NPC), but then after I got "stuck" as in no introductory quests at all.

I know this sounds very silly but I'd really appreciate if someone could give me a few tips... I'd really like to play this game.

Welcome!
There are not a whole lot of quests in old DAoC, and nothing in the way of a tutorial.
You can click on mobs to target them (or yourself), or use tab to target anything that is in range. The tilde key will initiate melee combat.
Sound like you started in Connla. There is a wandering sentinal there who you can '/whisper help' to to get kill tasks. You can do 1 task at level 1, 2 at level 2, etc. They will give you a good boost in experience and coin and make the grind shorter.
Focus on killing mobs with blue names. Yellow mobs are the same level as you, but particularly with the starter classes, you likely won't be able to kill them. Also, kill blue or green named humanoids whenever you can as they can drop cloth and leather armor which will make your life significantly less miserable for those first few levels.

Maybe goggle for DAoC tutorial? I'm sure there are introductory walk thrus out there. Probably even You tube vids to show you how things work.

Also, in game use /help and then select [The Basics]

You won't find many quests in DAoC anyway, be they introductional or not. You level by killing mobs, thats it. As Foigord said, better kill blue mobs until you found out how to do things.
In case you need help, you can ask in your starters guild with /gu. Also, you can ask in your group, if you have one. As you are in hibernia, feel free to pm me (or /t me) ingame whenever you need help, if I can help you, I will do so.

You might also want to take a look at the hibernia subforum, to find information about your new home ;D