How to unlock the great adventure achievement

At first, the credits available to you in the single player campaign is limited to that from missions alone. However, once you've fully maxed out either the zerg or protoss research trees, you gain the ability to trade any future research points into 10k worth of credits.

There have been mentions of a glitch with the way the game handles this: if you replay a mission you've already beaten and acquire protoss / zerg research points within that mission, it is possible to get the Moebius foundation to "purchase research samples" from you, even though you failed to increase your net points (you already got them once). In this way, you can accrue as many credits as you need, and can buy all the armory upgrades in a single playthrough.

Update: This method has since been patched.

As of now, the best way to earn the armory achievements is to play the campaign multiple times, selecting different upgrades on each playthrough, as achievements are shared across all campaigns.

According to the FAQ in Steam Discussions, Angels are gained by lifetime earnings:

Q: How do you get angel investors?

A: Your Angel Investor total directly correlates to your all time cash earned. To receive the bonus from your angel investors, you must redeem them by performing a reset to gain angel investors.

Q: Is there a formula for earning angel investors?

A: [150 * the square root of (your all time cash earned in quadrillions)] – AI spent. This means that when you earn an angel investor, it takes more money for you to earn your next one.