Been logging in to do dailies mostly with starter decks for about 3 weeks now and I buy a pack whenever I get 1000 gold to try and get more cards for my collection. Would it be better for me to save the gold to enter events / limited games?

It's most beneficial to play GRN draft. Easy to play (go boros) and best cards. With a bit of skill and luck you can chain drafts. Every deck is nowadays GRN based.

It seems to me like you should save gold for drafts, and save gems earned in draft for sealed, in terms of building collections. I could be wrong though.

On the other hand, I suspect when it comes to getting specific cards, you'll want to buy packs, if only to get more wildcards, since I don't think sealed and draft offer wildcards directly (you have to save up through the vault I think?). You probably won't build up your overall collection as fast, but you'll be possibly able to build a specific deck faster.

I think purchased packs offer less cards than a sealed or draft pack equivalent, something like 15 in sealed/draft vs. 6 in purchased, but most of the loss is in commons and uncommons, and the possibility of wild cards in purchased packs helps make up for it when building a specific deck.

So, if you only care about rares, mythics, and/or wildcards, purchasing packs directly seems fine, if you are more focused on building up your collection and do care about commons and uncommons more, draft/sealed seems the better prospect. Draft and sealed offer more rewards for winning in them, but are probably still good enough even if you lose every match to be better at building up commons and uncommons in your collection faster than buying packs directly with your gold.

I'd say play limited. There're two main reasons for this:

1) You're on equal footing with everyone else. You get to avoid getting stomped by people who simply have better decks than you in constructed.
2) You get to choose which card you want out of a pack (which is obviously better than random cards).

Also, for me at least, I find quick limited more fun than quick constructed. It's also more lucrative if one plays well + skillfully. Still, it's easy to earn back the entrance fee in quick constructed, so you can spam that too if you prefer.