There is a lot of model output available. Your decision about how to get the model results will depend quite a lot on what precisely you are after.
Other people are interested in these experiments as well, and could have already computed your diagnostics. This is extremely likely for the standard (DECK) experiments and future scenarios. There are several repositories where people are lodging these outputs.
There is expected to be 2-40 Petabytes of output uploaded as part of CMIP6, which requires substantial curation. The Earth System Grid Federation is a peer-to-peer network of data-stores to cope with all this data, mainly run as national facilities. There are instructions about how to access this data elswhere. There are multiple access methods such as:
Some of the experiments in PMIP phases 3 and 4 are not official CMIP experiments. This means that they may not be stored on the ESFG. Several of the groups have set up servers specifically to host these experiments (e.g. PlioMIP1 and DeepMIP). You will want to contact the leaders of those experiments directly to get access. This data may not be CMORized (put in a standard format with standard names).