How can I plot data with confidence intervals?

Here is a plotrix solution:

set.seed(0815)
x <- 1:10
F <- runif(10,1,2) 
L <- runif(10,0,1)
U <- runif(10,2,3)

require(plotrix)
plotCI(x, F, ui=U, li=L)

And here is a ggplot solution:

set.seed(0815)
df <- data.frame(x =1:10,
                 F =runif(10,1,2),
                 L =runif(10,0,1),
                 U =runif(10,2,3))

require(ggplot2)
ggplot(df, aes(x = x, y = F)) +
  geom_point(size = 4) +
  geom_errorbar(aes(ymax = U, ymin = L))

UPDATE: Here is a base solution to your edits:

set.seed(1234)
x <- rnorm(20)
df <- data.frame(x = x,
                 y = x + rnorm(20))

plot(y ~ x, data = df)

# model
mod <- lm(y ~ x, data = df)

# predicts + interval
newx <- seq(min(df$x), max(df$x), length.out=100)
preds <- predict(mod, newdata = data.frame(x=newx), 
                 interval = 'confidence')

# plot
plot(y ~ x, data = df, type = 'n')
# add fill
polygon(c(rev(newx), newx), c(rev(preds[ ,3]), preds[ ,2]), col = 'grey80', border = NA)
# model
abline(mod)
# intervals
lines(newx, preds[ ,3], lty = 'dashed', col = 'red')
lines(newx, preds[ ,2], lty = 'dashed', col = 'red')