You can substitute y = x + 3 and make y the subject, as you suggested, then substitute it back and solve for x.
You can use the balance method (basically "any operation is allowed, as long as you do it on both sides") to multiply by (x + 3) and get it out of the denominator.
Or you can write 1 = 1 / 1 and use either the balance method to "flip" both fractions (cf. a/b = c/d is equivalent with b/a = d/c as long as a, c are non-zero) or cross-multiply (cf. a/b = c/d is equivalent with a d = b c).
All of these lead to the same result (hopefully).
In general you want to take the x outside of fractions, brackets, etc. and then sweep them all to one side.