Wouldn't it be easier if you show us a screenshot of the champs you actually have and we could give recommendations who is able to do the dungeons. I mean, Urogrim is able to solo most dungeon content of the game. But what do you buy with that information, if you don't have that particular champ?

You're too early game to be able to solo Minotaur 15. Your priority at this stage is to build up a team that can be used across most basic content and especially Clan Boss. At this stage that means getting to 6* your starter champ, Ronda, and Deacon (if you got him), plus another that would fit well with them. Then progress more champs to 6* with the aim of being able to expand the content you can do and progress in it. :)

Pro-tip, you don't need to solo minotaur to do masteries.  i use 2 champs just to make it a much faster run, and the only thing i'm sacrificing is some XP for whoever I'm doing the masteries on.  still get the SOME xp, still get all the silver, still save ~200 gems.

If you can run 4 champs plus whoever you're mastering on mino 15, then do it.  Still better than spending 800 gems, getting 0 silver, and no XP.  FAIR WARNING, the entire team you put in needs their masteries done of course.  If you can't do mino15, probably better off just buying your masteries.

Some tips, whatever champ you picked as a starter, focus on leveling them up and giving them the best gear you can find, they will literally carry you into the mid game. 

from there when you have more strength, focus on Ronda. 

My personal pick for best Rare is Apothecary is a super strong support champion who is really easy to get to help your team. When you get some better gear, build him for Speed, HP and Defence, and focus on resistance if possible too for survivability and debuffs blocking. Again he's super easy to get and focus on.

Other great RARES, 

Frozen Banshee, Coldheart, Relequary tender, Harvester, Bulwark, Shadowbow Tirlac, painsmith, and Crimson Slayer 

are all good rare champions to help you through varrious bosses and progression.

As for epics, there are a ton, 

Aox the rememberer, Mausoleum Mage, runekeeper Dazdurk, Geomancer, Deacon Armstrong, godseeker Aniri, Seer, Sachi, Skullcrusher, Vrask, high Khatun, Jareg, Stag knight, Tayrel, steelskull, Skiuremis, Fenshi, Whisper, Prundar, Seeker, Miscreated Monster, quargan the crowned, Uugo, Gogorab, Shaman, Psylar, Akemtum, Alure, Sinesha, Genbo, Toragi, Daithi, Orn, Demytha, Morag, Grizzled Jarl, Madame Seris, Skullcrown, Gorlos, Skimfos, duhr, Survivor, Zargala, Skraank, skytouched shaman, Juliana, Fenax, Battlesage, Andryssia, Eolfrig, and Lady Annabelle 

All of those if you pull any will be considerable help to some part of the game. 

I hope this was a bit more imformative as to answer your question directly. Those 9 rares especially are extreamly above their rarity power levels and will help you well enough. The epics listed are some of the best 3.5+ rated epics that will help you mid to late game if you can pull them. If you do pull any of them, I suggest just looking up who they are an reviewing their skill kits to see why they are so good an at what parts of the game they are best in. 

Best of luck to you!