How can I most effectively farm Pixels?

You can use the refinery to turn excess ore into pixels fairly effectively. To get this, you will need the tier 2 crafting station, which you can only access after defeating the first boss. This is, however, a very effective way to get extra bits out of excess ore.

Exploring the surface for large buildings can prove quite fruitful, as many of the chests and containers therein will contain large caches of pixels. Take them. All of them. As of right now (shortly after the Frustrated Koala update), NPCs don't even get mad if you take their stuff from their containers.

If you cant' find such structures easily, there's another way to find large pixel caches.

Dig deeper down.

The world I am on has a huge underground cavern that has tons and TONS of capsules full of pixels. dig down deep enough, and you'll start running into them all over the place.

This area is dangerous though, so be prepared.

For extra bits, make sure you kill monsters with a strong Melee weapon, which provides a high chance of them dropping bits (with ranged weapons, the chance for bits to drop is lower due to a higher chance to drop meat and leather instead).

Search for a Sewer, but don't kill the po golems , search the whole sewer and lead all the po golems to one spot.

If the po golems can't hit you, they will start throwing small versions of The po Dig a 4/6 wide gab between you and the po golems so the small po's fall in. The small po's can be killed easy and you can collect the pixels quite easy!

Make sure the range attacks of the golems can't hit you by placing one block in front of their face