This doesn't really have much to do with minfi, but instead has to do with how you specify/interpret linear models in R. In other words, minfi is doing special things to smooth data across CpG sites, but when you fit the model, you are just fitting a linear model on data, using R.
While you can often get answers to explicit modeling questions on this forum, in general the expectation (for anybody who is analyzing data, really) is that the analyst knows what he/she is doing. Unfortunately, the aspects of linear modeling, plus how models are implemented in R, is something that is WAY beyond the scope of a support forum. Linear modeling alone is like four or five of the courses I took to get my master's, and that didn't include figuring out how to do these things in R (back when I was in school, we used this nasty thing called SAS, which makes me shudder to even think about...).
Anyway, there is no substitute for knowing what you are doing, and you aren't going to get that here. Good resources are the Users Guides for limma and edgeR, the DESeq/DESeq2 vignettes, which have lots of examples. Julian Faraway's book is also a really good resource. Plus I am sure you can dig up other things using the googles.