There are a large variety of experiments you can choose to measure various epigenomics marks. Histone modifications, which are associated with a variety of chromatin states like promoters, enhancers, active transcription, and silenced transcription, can be measured by ChIP-seq.
Regions of open chromatin (i.e., absence of nucleosomes), which are associated with active transcription and protein-DNA binding, can be measured by DNase-seq or FAIRE-seq. Alternatively, nucleosome positioning can be measured by MNase-seq. However, a new methodology, ATAC-seq, can be used to measure both open chromatin and nucleosome positioning (the latter only if paired-end sequencing is done), and is an easier protocol to follow.
DNA methylation can be measured by MeDIP-seq or bisulfite-seq (BS-seq). MeDIP uses an antibody to pull down methylated regions, and thus gives a broad measure of DNA methylation in a gene locus. BS-seq relies on chemical conversion of non-methylated nucleotides, and thus gives single-nucleotide resolution of methylated DNA, at the risk of false-positives due to incomplete conversion.
Finally, long-range looping interactions consistent with enhancer-promoter regulation or long-scale DNA structure can measured by the chromosome conformation capture family of methodologies (3C, 4C, 5C, Hi-C). These protocols can also be linked to an immunoprecipitation step to measure looping in the context of a specific protein (CHIA-PET).