I don't really get why anybody plans any IPO prices. Shouldn't auctions be the best tool for the current owners to get as much money as they can?

They actually are auctions, just not public ones. Silent auctions, where the bidders communicate privately with the seller.

The buyers set a price they will pay, possibly a price x number of shares matrix.

Then the company picks a price to maximize the amount brought in.

But why did the share price then go up on the first day after the IPO? (And why does it tend to do this for a lot of companies?)

Yes, more or less. But you often see shares go up by quote a lot on their first day. That should not happen.