Ok, I can't for the life of me catch the bird's with my bug net to get feathers. Is there some trick to it? Semi related, is it worth upgrading my Sacred Shield (which I need the feathers for) or is a replacement shield right around the corner (I am currently getting the pickup for the Water Dragon).

Creep up slowly behind them, with your wiimote pointing downwards and tilted to the side to make a nice sweep. I didn't try birdcatching before getting the net upgrade though, so that may be a must.

upgrading your sacred shiled is a very good idea. You won't get the hylian shield until very late in the game. Pay attention that it doesn't break, though, you're going to be very pissed if your hard-earned upgraded shield breaks. For lvl 3 sacred shield you're going to need 3 or 4 dark half moons, so don't waste them until you have it.

As for the bird catching, yeah it's tough. Slowly walk towards them and then get them with a fast horizontal swipe, this is how I had the best chances getting one. Keep in mind that you get blue feathers from blue birds while all other birds give you normal feathers.

I read somewhere that you can heal/fix your shield with items.

Yes, but under the condition that it isn't completely broken. Also, this is only interesting for your wood or iron shield, as the sacred heals from itself.