There are a few different things that could be looked at. Firstly, assuming you ran control samples from the dogs in addition to the cancer samples, the first thing to do would be to perform standard differential expression/methylation analysis. For DE, the edgeR, DESeq2 and limma packages are very good and what you'll find everyone recommending. Note that I'm not sure how good the annotations are for the dog genome (I don't work on it), so you might need to use something like RSEM (or trinity followed by RSEM) to get decent metrics, which means you'd be stuck with limma downstream (not that that's a bad thing, limma is an extremely powerful tool). For methylation, it depends on how you generated the data. For RRBS or similar datasets, BiSeq is OK. For methylation arrays, you can use packages like minfi or COHCAP.
One of the interesting things I would do is use GSEA to compare enrichment of groups of differentially expressed/methylated genes between the canine model and patients. You'll obviously need control patient data for this to be worthwhile. If you find any highly relevant pathways (there are a few bioconductor packages for pathway analysis, though I think the Ingenuity Pathway Analysis commercial package is still better in this regard) then I'd pay particular attention to how key players in them are affected in patients.
That's a quick idea and a handful of Bioconductor packages to get you started. I could probably come up with things to look at all day, you have a really target-rich project :)
Not Sure about the bioconductor package. But the way, I would do is.
1. I would map RNA-seq reads on the genomes of both species and would fetch common regions where reads are uniquely mapped. If these regions are supported by other reads then they are orthologous regions getting transcribed.
2. To get Phast conservations scores of DNA methylated regions.
3. Once I have these informations then I can play around on R.