Future of CMB observations: How will our knowledge of the early universe change?
This is a great question. I know of a couple of really big things about inflation people want to be able to nail down by using the cosmic microwave background. 

The first is measuring what are known as E- and B- modes, which are the curl-free and divergence-free components to the modes of cmb radiation:



Essentially, measuring large scale Gaussian B-modes from primordial gravitational waves will help constrain the energy scale of inflation. It may also be able to rule out most ekpyrotic and pure curvaton/inhomogeneous reheating models (same source).

The other thing people are looking at is this idea of primordial non-Gaussianity, which are second order corrections to the Gaussian fluctuations present in the cmb (review article; early planck results). Measuring a parameter called $f_{nl}$ (deviation from Gaussianity) has been a fairly crucial part of current and future studies and will also help rule out various inflationary models. This $f_{nl}$ parameter is defined as follows:


  
    In this case the multipole coeﬃcients $a_{lm}$ of the CMB
    temperature map can be written as
    $$a_{lm} = a_{lm}^{(G)} + f_{nl} a_{lm}^{(NG)} $$
    where $a_{lm}^{(G)}$ is the Gaussian contribution and $a_{lm}^{(NG)}$ is the non-Gaussian contribution.