"I am going by bus with my mom."
Translation:Jadę autobusem z moją mamą.
I'm sorry, but I disagree. It is logical to assume that I am going with my own mom, and if it's someone else's mom, then I specify it. This particular sentence doesn't sound so well to my ear without the possessive, but it's surely correct.
I added the possibility to omit the possessive.
Forms of "iść" and "chodzić" are either for movement on foot (so simply walking), or when the means by which you go somewhere are irrelevant. For example "Jutro idę do kina" = "Tomorrow I am going to the cinema" does not necessarily mean that I will walk there, it's just that it really doesn't matter that I will take a bus.
Forms of "jechać" and "jeździć" mean that you are using a vehicle. A bike, a bus, a car, etc. But only the land ones (which presumably have wheels), no planes or boats.
So as the bus was explicitly mentioned, "idę" is not an option here.
One note about the 'irrelevant' part: I guess it doesn't work when it's hard to imagine someone could really walk to the given destination. Going to another city (or even city district) or abroad almost definitely implies some vehicle used, it's just too far, and if someone really wants to walk, well, that's a pretty non-standard situation.