Treat your data as time series and the solution comes very easy:
df <- read.table(text="Year AL AK AZ AR CA CO
1993 135 153 113 157 718 516
1994 218 154 184 185 845 465
1995 482 846 683 682 863 863", header=T)
time.series <- ts(df[, -1], start=1993, end=1995)
plot(time.series, main="Toy example")
which produces
If you want a single panel, then:
plot(time.series, main="Toy example", plot.type="single", col=1:6)
You may want to read ?legend
to get know how to put legends on plots.