There are two key steps to arranging the data so that the plot is easily made. You should set the column containing dates to have date values instead of character strings and you should reshape the data so that a single column contains the region labels and a single column contains the values to be plotted. I do this in the code below, using the name of US states rather than your (Norwegian?) regions. Notice that the states labeled as New_York and New_Mexico in the data are listed in the legend without the underscore because I manually entered the legend text.
DF <- data.frame(kvartal = c("2011-03-01","2011-06-01","2011-09-01","2011-12-01",
"2012-03-01","2012-06-01","2012-09-01","2012-12-01"),
New_York = 1:8 + rnorm(8,0,1),
Florida = 2:9 + rnorm(8, 0, 1.5),
Texas = 1.5:8.5,
New_Mexico = 0:7 + rnorm(8, 0, 0.5))
DF
#> kvartal New_York Florida Texas New_Mexico
#> 1 2011-03-01 0.04125076 0.2634613 1.5 -0.2976835
#> 2 2011-06-01 3.96233379 3.1024347 2.5 0.6971117
#> 3 2011-09-01 5.07298595 3.5399068 3.5 1.2338146
#> 4 2011-12-01 4.59675228 3.2087747 4.5 3.0602612
#> 5 2012-03-01 5.42429997 6.8475503 5.5 4.2669195
#> 6 2012-06-01 6.28974342 6.5363706 6.5 5.5786714
#> 7 2012-09-01 7.83045386 7.4183318 7.5 5.6547064
#> 8 2012-12-01 8.88240002 7.9082453 8.5 7.0834392
library(tidyr)
library(lubridate)
#>
#> Attaching package: 'lubridate'
#> The following objects are masked from 'package:base':
#>
#> date, intersect, setdiff, union
library(ggplot2)
#Make the kvartal column a date value
DF$kvartal <- ymd(DF$kvartal)
#pivot the data to have a column labeled Fylke with the state names
#and a single value column
DF_long <- DF %>% pivot_longer(New_York:New_Mexico, names_to = "Fylke", values_to = "Value")
head(DF_long)
#> # A tibble: 6 x 3
#> kvartal Fylke Value
#> <date> <chr> <dbl>
#> 1 2011-03-01 New_York 0.0413
#> 2 2011-03-01 Florida 0.263
#> 3 2011-03-01 Texas 1.5
#> 4 2011-03-01 New_Mexico -0.298
#> 5 2011-06-01 New_York 3.96
#> 6 2011-06-01 Florida 3.10
ggplot(DF_long, aes(x = kvartal, y = Value, color = Fylke, group = Fylke)) + geom_line() +
labs(x = "Year", y = "Antall arbeidsledige") +
scale_color_hue(labels = c("Florida", "New York", "New Mexico", "Texas"))

Created on 2020-05-06 by the reprex package (v0.3.0)