I seem to recall hearing that you are supposed to win better cards by beating opponents, but right now that seems a tall order if not impossible. All the opponents I run into have powerful cards, heroes, special abilities etc., and I just have a few basic low-strength cards. Am I supposed to just put the game aside until I can buy/find better cards, or is there some kind of strategy I'm missing? Thanks.

You can win with the starter deck, it's what people all over the globe do. Look online; there are plenty of guides on how to utilise your deck properly. Remember: practice makes perfect, so do it, don't just read about it.

Well you obviously have to play against the worst players at first. I guess that's what you are doing wrong. If the opponents you play against have such strong decks, you select the wrong players.

It is easy even on hard gwent difficulty.

look at this map and locations: http://witcher3map.com/v/#4/-79.88/67.10/m=-75.500,110.810

go around buy cards, play with this guys on the map,
you get better cards, than play with stronger players and do gwent quests.

learn how to play, that is your main problem and solution.

Once you are in Velen go to heart of stone part of the map, the win mill with two black rune merchants, you can buy nice cards there with some leaders, also go at caravan near big mansion where you go to party mission with triss, guy there sell good cards and play him as well. That way you got very strong cards, gunter of darkness and nice leaders for every deck. Also party location with ghost and medic red heard shani girl, in the barn on the leaders there is cow card, strong card as well.

Just google all this stuff and you will find it easy, than start playing stronger players.

The first few were easy, but when I reached the nobleman in the palace courtyard (right after talking to the emperor and Yen) it suddenly got a lot harder. He was pulling out heroes and strength 10 cards. I think I tried over a dozen times to beat him and couldn't do it. If all opponents will be this hard from now on, I'm not sure what to do.

Invoke the 'pawn' gambit: give him the round. You must learn how to tactically lose rounds as well as win them.

Yeah, it can be a little frustrating at first, because some opponents pull some straight magic on almost certain victories. Look around at the vendors. Some sell cards that can come in handy. You also want to play your deck to your advantage. The Northern Realms deck and Leader Ability are strong against Scoia'tael decks, but you'll probably have a hard time against Nilfgaard.

That guy actually is a special difficulty spike. He's much, much harder than your average Velen-area player.

Just buy up nice cards when you see them and you'll do fine. Go back to that guy in the Vizima courtyard later, maybe after you do Velen and Skellige, and you'll have almost no trouble with him.

That makes sense then. I'll go back to him later, thanks!

Nah, it's not frustrating, it's bad design. You don't present players will challenges they are not able to master without giving them a hint they're not supposed to win.

That's just terrible design and a huge waste of time.