0.59999... is, in fact, 0.6 and 0.6=6/10=3/5 as a fraction.
0.244444... is a bit trickier. First, multiply by 10 and we get 2.44444...
To convert the recurring 4 into a fraction we use as many 9's as the length of the recurring part. Well, there's only one digit recurring, so we only need one 9; make a fraction of the recurring part and the 9: 4/9.
Nearly there. We had 2.44444... so that's 2 4/9. Express this is an improper fraction: 22/9 and now we must divide by 10 to compensate for multiplying by 10 earlier: 22/(9*10). This cancels a bit: 11/45, which is the fraction you're looking for.
(We can use the technique in the preceding paragraph to the first part, 0.599999... Multiply by 10 first: 5.999999, then take 9 and divide by 9: 9/9=1 so we have 5 9/9=5+1=6. Then we divide by 10: 6/10 and we get 3/5.)