There is a database called OmicsDI, where one can search for multi-omics datasets.
Here's a link of the associated publication (Perez-Riverol, Yasset, et al. "Omics Discovery Index-Discovering and Linking Public Omics Datasets." bioRxiv (2016): 049205.) for more details.
In general, there is two options to identify targets for transcription factors: experimental (ChIP-seq) and sequence-based predictions.
TF binding from experimental data
The are multiple projects that produce binding data of transcription factors and quantify their peaks across the genome. The advantage here is that you know that binding actually occurs, ...
iRegulon takes a sequence-based approach to finding transcription factor targets. There's a Cytoscape app that you can use to find the regulators of a given gene list, or the targets of a particular transcription factor.
Transcription factor binding sites are predicted using a collection of position weight matrices (PWMs) from a number of sources, including ...
There are several datasets available on GEO, though you do have to search for them. For example, here are three data sets that have both Illumina methylation and gene expression microarray profiling:
Next to OmicsDI the EBI has a special repository for multi-omics datasets: https://www.ebi.ac.uk/biosamples
It links the different datasets between repositories, ie. PRIDE for MS/MS based data and ArrayExpress for RNASeq data.
The steps you describe are correct. For step 2 it is usually normalized to mean 0 and variance 1. However the "machine learning" part is important.
Having several samples being technical replicates will make the integration task easier. However, you have too few samples to make any good prediction. At most I would describe it as an exploratory analysis.
I suggest you give DAVID a try. Specifically, their Functional Annotation tool. Just enter your list of protein IDs, and it will return groups of proteins where particular GO functions are overrepresented.
This should at least help you start categorizing your list into functional groups.
I tried dada2 and is not bad (if you know R).
QIIME2 is also an option.
Many other are available, the choice might also depend on your sample and your exact question.
For functional profiling you may try picrust2
Welcome to Bioinformatics Stackexchange @SD1024. A recently published algorithm known as MOFA (Multi-Omics Factor Analysis, paper, github) is generating a lot of interest, and is designed answer exactly the sort of biological question you are describing. It claims to extract axes from multiple matrices with overlapping samples, but not necessarily ...