On a basic level:

W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574–592.

L. Ahlfors, Conformal invariants, last chapter.

S. Donaldson, Riemann surfaces, Oxford, 2011. Very nice. Modern.

R. Courant, Function theory (if you read German or Russian, this is the second part of the famous old Hurwitz-Courant textbook, not available in English).

On even more basic level:

G. M. Goluzin, Geometric theory of functions of a complex variable, AMS 1969,
Appendix.

(It depends on the definition of the Riemann surface that you are willing to accept. If you want to include the triangulability in the definition then Goluzin is fine, and this is probably the simplest proof available. Triangulability is
equivalent to the existence of a countable basis of topology, which is not logically necessary to include in the definition (it follows from the modern definition of a RS, but this fact is not trivial). On the other hand, I know of no context where Riemann surfaces arise and the existence of a countable basis is in question.
The proofs in Ahlfors and Abikoff include the proof of the existence of a countable basis, and do not discuss triangulability.)

Let me also mention a remarkable recent book:

Henri Paul de Saint-Gervais,
Uniformisation des surfaces de Riemann, ENS Éditions, Lyon, 2010. 544 pp.
There is an English translation published by EMS.

This is not for quick reading, but it contains a very comprehensive discussion of the history of this theorem, and early attempts to prove it, and various approaches, etc.

EDIT. Let me mention a new book, it is actually a textbook for graduate/undergraduate US students which contains a complete proof:

Donald Marshall, Complex Analysis, Cambridge 2019.

This seems to be the unique general CV textbook which contains a complete proof.