do you have to spend money in order to get certain weapons?
Not at all, everything's craftable or buyable for currencies you can obtain just by playing.
how long will it take to gain loot without spending money?
Depends what loot we're talking about.
Getting Ash Prime, Latron Wrath, and Gaia's Tragedy will take a long time. But getting just upgrades to your current gear? Nah, it mostly is a breeze, as long as you keep unlocking new planets.
what should my priorities be as a brand new player?
what weapons/warframes/etc are best for starting players?
Can the game be played start to finish completely solo?
Can it be? Yes, it can.
Will it be easy? Fuck no.
Until you get a good collection of high-level mods, high-tier weapons and frames, doing most missions - especially endless ones - will be a royal pain. You will be dying. You will fail those missions. You will get frustrated after 10th failed attempt.
what is the best way to farm materials?
For common and uncommon resources, endless missions, especially Dark Sector ones. If you have Nekros or Hydroid in your team, it's even better.
For rare resources, killing bosses is usually a good idea. Especially iff there's a Phorid Manifestation on the boss node.
how long will it take to unlock additional warframes?
You can unlock Rhino as early as Venus, so not long at all.