I just wanted to chime in to plug my own Ext calculator (a joint venture with Dexter Chua). You can see an interactive calculation of $\mathbb{R}\mathrm{P}^4\wedge\mathbb{R}\mathrm{P}^4$ here:
You can interactively add Adams differentials and it will propagate them using a few basic rules, but you'll mostly need to work them out yourself.