Dive!

library(tidyr)
library(data.table)

Part 1

dt <- data.table(read.table("input.txt", sep = "\n"))
dt <- separate(dt, "V1", c("d", "n"), sep = " ", convert = TRUE)
dt$n <- ifelse(dt$d == "down", dt$n * -1, dt$n)

abs(
  sum(dt$n[dt$d == 'forward']) 
  * as.numeric(sum(dt$n[dt$d != 'forward']))
)

[1] 1635930

Part 2

dt <- data.table(read.table("input.txt", sep = "\n"))
dt <- separate(dt, "V1", c("d", "n"), sep = " ", convert = TRUE)

dt$aim <- ifelse(dt$d == "up" & dt$d != "forward", dt$n * -1, dt$n)
dt$aim <- ifelse(dt$d == "forward", 0, dt$aim)
dt$aim <- cumsum(dt$aim)

dt$h <- ifelse(dt$d == "forward", dt$n, 0)
dt$depth <- ifelse(dt$d == "forward", dt$aim * dt$n, 0)

sum(dt$h, na.rm = TRUE) * sum(dt$depth, na.rm = TRUE)

[1] 1781819478

Speed Edition

Base R once again beats out data.table by just a little bit. The big difference here was moving from colSums to sum, although readLines instead of fread does play a part.

Fastest solution

dtM <- data.frame("V1" = readLines("input.txt"))
dtM <- tidyr::separate(dtM, "V1", c("d", "n"), sep = " ", convert = TRUE)

# part 1
dt <- dtM
dt$n <- ifelse(dt$d == "down", dt$n * -1, dt$n)
abs(sum(dt$n[dt$d == "forward"]) * sum(dt$n[dt$d != "forward"]))

[1] 1635930

# part 2
dt <- dtM
dt$aim <- cumsum(ifelse(dt$d == "up", dt$n * -1, 
                        ifelse(dt$d == "forward", 0, dt$n)))
sum(ifelse(dt$d == "forward", dt$n, 0), na.rm = TRUE) *
  sum(ifelse(dt$d == "forward", dt$aim * dt$n, 0), na.rm = TRUE)

[1] 1781819478