What is the best approach to calculate the transfer function of this RC circuit?
Mesh analysis
Nodal analysis
Stepwise mesh-then-nodal, working backwards.  This only works for ladder circuits like this, but it's kinda fun:


Assume everything is in the Laplace domain.  I should have capitalized my signal names, but I didn't, I only just realized it, and I'm lazy:


Label a \$v_2\$ and \$v_1\$ in the obvious spots.
Assign currents \$i_1\$, \$i_2\$, etc., to \$R_1\$, \$R_2\$, etc.
Calculate \$i_3\$ as a function of \$v_3\$.
Calculate \$v_2\$ as a function of \$v_3\$ and \$i_3\$.
Calculate \$i_2\$ as a function of \$i_3\$ and \$v_2\$.
Etc., until you get back to \$v_i\$
Now start substituting like mad (in simple steps, which is why I like it) until you get the great big nasty formula for \$v_i\$ as a function of \$v_3\$.  Then do just a teeny bit more algebra so that you have \$\frac{V_3}{V_i}\$.