Is it correct to assume that going fast is more about riding the fins properly?
If this assumption is correct, does that mean my board does not necessarily need to be top of the line, but more importantly that my fins are?

Great fins mismatched to a crap board will likely still be crap, if the board is not a modern 190 x 70 it will likely be over finned, (throw you over leeward edge) unstable (hard to keep flat) and or difficult to keep the board going straight without a ton of foot pressure if using the same fins as the fastest guys. If the fin builder is familiar with the board you plan to use and has a solid recommendation it could haul ass. Otherwise you're in for some trial and error to get it dialed. Tough to start out learning to ride a raceboard if you're gear isn't good to start, hard to tell if it's you, the gear, or both.

So the board is 189 X 59. Would like to buy good fins for it, then later, when budget permits, buy larger board. Then take good fins from older board and put on new board.
Is this NOT a good option?

No, caus most likely the two different boards need two different sets of fins.

But if you go for a 1 1/2 year old tri-fin raceboard you could most likely could use the fins on a new as well.

But just to come with an example of why not only fins are important. If eg. Just the fin boxes are not mount with the right distance/balance in the board. Even a board with the right dimensions and fins would be crap.

this so true.

I just got a new set of fins, and it completely changed the board. I was both amazed and surprised at the difference. Especially big difference at higher speeds. More controllable and and much easier to ride. Good times.

If you decide to put a set of quad fins on the 183, Patrick Rynne describes a set of quad fins that worked well on it.
viewtopic.php?f=1&t=2372262&p=716730&hilit=fins#p716730

They would only suit the 183, which I am guessing is what your board actually is?

Also, the new boards are Tuttle Box and many of the 59cm boards were Power Box.

I purchased some used Rista fins for my 183x59 Cabrinha board. Quad fin set 33.5cm rear and 37.5cm front. The set was originally for a 65cm RRD board, but I figured I'd give it a try. The fins are oversized. OK to ride in light winds though. Goes upwind like crazy compared to the old fins, but I get overpowered very quickly and need a ton of foot pressure to keep the board flat.

Still on the lookout for a used tri-fin 70cm board. I'm hoping all those olympic windsurfers will want to dump their 6 month old gear they bought during kiteboarding's brief time as an Olympic sport