Edited according to new information OP gave. I'm not a CS major so someone else could provide a better answer.
In my knowledge on the field of CS, I believe that most universities will give you full tuition waiver and living stipend, and you will be expected to work as RA or TA along the way. I do know some of the last year PhDs who taught intro class in undergrad CS as instructors.
I've been led to believe that it's much easier to get money (assistantships and fellowships) as a PhD student than as a master's student
PhD students are more likely to be under assistantships and fellowships, and most likely it may be required for them to be under assistantships or fellowships. Some excellent Master's student may also get such opportunities. However, if you are referring to opportunity cost, the answer becomes more of personal preference than an absolute answer. If you know that you ultimately want to do PhD, it is financially better to go straight into PhD without having to pay for your tuition for Masters. However, going through masters may provide a better chance for your admission acceptance. You will also be more sure about your research interest; jumping into PhD is a huge investment, so it's wise to explore before committing. There are many CS majors that I know personally who went on to MS before going into PhD.
And are there other significant factors in determining whether or not it's cheaper to go straight to a PhD program?
I think ultimately you have to check with the graduate programs about their funding. There are many Master's programs in CS that provide partial tuition waiver, and if you become a TA, they may even waive all your tuition in some universities. Look into your universities to see if they have such programs available.
If finance is one of your concern, the advice that I've heard is to work after undergraduate in research institutions or industry. In that way, you can earn some money for yourself, while boosting your resume and giving time for yourself to discover if you like the research topic you were initially interested in. If you end up in an industry job with little research aspect, publishing papers along the way is excellent way to show admission committee that you are interested in academia.