In a nutshell, wedges and a driver.
First, I bought two L38 ATV wedges, a 60 and a 64. I considered the Clevelands, and I'm sure they're fine, but I liked the distances a little better on the 64 TaylorMades (50 instead of 60). They've made a HUGE difference. Even with mediocre balls, they throw darts at the green. And high winds have almost no effect on them. At a minimum, get 2 wedges to replace the sand and lob that come with your starter set.
And today, just after I hit level 58, I bought the G20 driver. I was really struggling with longer holes after turning Tour Pro. I considered the K15 because of it's price and slow meter speed, and at 250 yards would have been a huge upgrade over the starter, but I wanted that extra 20 yards from the G20. It will carry me much further in the game, so I didn't see the point in spending 1100 on the K15 when the G20 is just 400 more.
I just played my first round with both the wedges and the driver, and it was like I was playing a different course. On the
first hole at St. Andrews, I hit the new driver, then dropped a 60 degree
wedge shot within a few feet and drained it for a bird. I shot 67, my best round to date. And that included a couple of putts that I should have made, but I'm still somewhat timid on the faster greens after blasting a few putts 25 feet past the hole in a recent tournament.
Anyhoo, if you're still using the starter set, consider upgrading these 3 clubs. I know, they cost "money", but I got around that by buying 2 domain names at Go Daddy for $3.99 each. They have to be your first domain purchases, but for $8, I got 3,000 credits, which would normally cost $30.
Now, if I could only find an affordable golf ball that performs well ...