Firstly, when looking at costs, I find a lot of companies don't actually understand fully the cost of operating their current kit. They then often try and do an Azure/IaaS/SaaS price comparison by just layering their existing topology into Azure (other vendors are available). Both of these lead to a difficult business case.
YFirst, you really do need a solid understanding of your current real costs so that you know what you're comparing. Next, to look at the hosting there are typically two stages to examine - take on (or lift & shift as you've referred to it above) and then optimisation. People call it different things, transition and transformation for example.