Please take this page in conjunction with the Part III Guide to Courses Statistics section.
Please read in particular Introduction and prerequisites given in the Guide to Courses.
Essential, helpful or helpful for some courses as listed in the prerequisites page linked to above.

Basic Statistics (below)

(Part II Probability and Measure: see Measure Theory)
Basic Statistics
You will need this for the following Part III courses:

Modern Statistical Methods

Statistical Learning in Practice

Statistics in Medicine

Causal Inference

Bayesian Modelling and Computation

Topics in Statistical Theory

Astrostatistics
Relevant undergraduate courses are:
First level prerequisites

Basic probability theory, including common distributions such as Binomial, Normal, Poisson, Chisquared, etc. See e.g. Basic Probability and official prerequisites page mentioned above.

Basic Linear Algebra. You can check if you are at the required level by doing the following exercises: Statistical Modelling Example Sheet 1 (2014/15) Q 1, 2a, 5, Example Sheet 2 (2014/15) Q8a

Basic Real Analysis. You can check if you are at the required level by doing the following exercises: Past Exam Question Part IA 2016 Paper 1 Q3, Part IA 2012 Paper 1 Q3
Second level prerequisites
Undergraduate statistics, including:

Maximum Likelihood Estimation. Common examples, consistency and asymptotic normality.

Hypothesis Tests and Confidence Intervals. Likelihood ratio statistics, Wilks' theorem, ttests and Ftests.

The Linear Model. MLEs, hypothesis tests, confidence intervals and extensions.

Bayesian Statistics. Definitions, examples and comparisons with MLEs.
You can use the following exercises to check your present level and to guide your study.

Part IB Statistics Example Sheet 1 (2014/15) Q3, 8, 9, 10

Part IB Statistics Example Sheet 2 (2014/15) Q1, 2

Part IB Statistics Example Sheet 3 (2014/15) Q1, 4, 5, 9
Useful books and resources

IB Statistics course page by Prof. Spiegelhalter with notes and relevant information.
Basic Probability
The relevant Cambridge undergraduate course is Ia Probability.
Useful books and resources

Prof. Weber's lecture notes from 2015.

More very good resources on Prof Weber's page.
Markov Chains
Relevant Cambridge undergraduate courses are IB Markov Chains.
Useful books and resources
 Parts of a book by Prof. Norris.