An alternative approach (not necessarily better than the previous answers!) is to use the viridis package. As explained here, it allows for a variety of color gradients that are based on more than two colors.
The package is pretty easy to use - you just need to replace the ggplot2 scale fill function (e.g., scale_fill_gradient(low = "skyblue", high = "dodgerblue4")) with the equivalent viridis function.
So, change the code for this plot:
ggplot(mtcars, aes(wt*1000, mpg)) +
  geom_point(size = 4, aes(colour = hp)) +
  xlab("Weight (pounds)") + ylab("Miles per gallon (MPG)") + labs(color='Horse power') +
  scale_x_continuous(limits = c(1000, 6000), 
                     breaks = c(seq(1000,6000,1000)), 
                     labels = c("1,000", "2,000", "3,000", "4,000", "5,000", "6,000")) + 
  scale_fill_gradient(low = "skyblue", high = "dodgerblue4") +
  theme_classic()
Which produces:

To this, which uses viridis:
ggplot(mtcars, aes(wt*1000, mpg)) +
  geom_point(size = 4, aes(colour = factor(cyl))) +
  xlab("Weight (pounds)") + ylab("Miles per gallon (MPG)") + labs(color='Number\nof cylinders') +
  scale_x_continuous(limits = c(1000, 6000), 
                     breaks = c(seq(1000,6000,1000)), 
                     labels = c("1,000", "2,000", "3,000", "4,000", "5,000", "6,000")) + 
  scale_color_viridis(discrete = TRUE) +
  theme_classic()
The only difference is in the second to last line: scale_color_viridis(discrete = TRUE).
This is the plot that is produced using viridis:

Hoping someone finds this useful, as its the solution I ended up using after coming to this question.