However, I don't see why I shouldn't work towards another.
Warframes, unlike most games, aren't like set classes. While their abilities and powers favor certain playstyles, most are exceedingly versatile, and can be built in any number of ways. Rhino isn't necessarily a tank; he can just take a lot more hits than others. You can boost his speed to make him generally on par with other frames, decide you want to up his tankiness even further, or focus on his ability power.
There are 22 Warframes; there's totally room to experiment! And your favorite should always still be available if you don't prefer the others!
Should I just "pointlessly" grind through the planets to get new ones which will hopefully carry what I need?
A much better strategy would be to use the myriad of resources at your disposal to learn about the game! The Warframe Wiki is by far the most comprehensive: http://warframe.wikia.com/wiki/WARFRAME_Wiki
Search each of the planets, and it'll tell you what drops from there. Then just go there and start farming! Easy peasy. Our FAQ also has a guide that details specific spots players like to farm each resource from: http://www.reddit.com/r/Warframe/wiki/faq#wiki_where_is_the_best_place_to_farm_a_specific_resource.3F
I wanna get a Cernos, Paris/ParisPrime, and Dread.
Paris and Cernos are definitely the most attainable for you right now. I would hold off on Dread until you can reliably fight Stalker, and I would definitely hold off on Paris Prime until you're more familiar with the game, the missions, and how to use Void Keys.
in terms of quest progression; What exactly am I looking at in the future? If the Kubrow and Archwing quests are more lategame, am I literally to just find my own goals now (the game is "over") or is there something I should be looking into?
Like I said before, there isn't a set "progression" in the game. Rather, it's how far you can push yourself. If you want to farm a certain item for example, then go do it! If you want to push through enemies at a higher difficulty than you might think, and succeed, then you can reap the rewards. If not, farm up on the missions you can beat, and rise in power to match those challenges.
Having your own goals doesn't mean the game is "over." The meat of the game is the ridiculous amount of weapons and Warframes available to craft. While your interest right now may be in Volt and the bows, I would encourage you to branch out and try some of the other weapons and gear that interests you. I recommend the Penta when you're able to get the resources for it :D
What about bosses? Should I be looking into killing them?
Yup! Each boss unlocks the adjacent planets, so expanding your reach in the galaxy by killing them is prudent. In addition, each boss drops the parts blueprints for a certain Warframe, making them quite useful to earn new gear as well!
I guess it just feels weird that I finished just a couple of quests and now there's seemingly nothing other than grind. Other games' storylines/quests tend to last more than 2-3 hours.
Warframe isn't like an MMO or a set game. It's more freeform than that. It's a lot of freedom, to basically create whatever gear you want and amass your space-ninja arsenal. For some folks, it's too much freedom, and it feels like there's no content, but there's a difference.
Sometimes content is having things very structured, from item progression to story. Other games are more fluid in their content, presenting content as open for earning and story in chunks. Every game has its groove, and Warframe has a unique one!