I guess dungeons are optional but looking for ore veins isn’t and given the probability of some veins, chances are you’ll find plenty of dungeons either way. That said chrome is totally accessible in early game (basalt vein with reasonably probability) it requires a quality 2 tool to mine and you can process it inefficiently in a crucible for early game chrome. Manganese is tough in the early game and dungeons are honestly your best bet ,other methods require infrastructure. If you however refuse to go dungeoning or have disabled them i suppose, you can’t produce gray vitriol because you can’t bath ores in sulfuric acid without a stainless steel bath and you cant produce sulfuric acid either i think, so you have the following options: Make a sluice to get sluice juice -> dry the sluice juice to get sluice sand -> make a low tier magnetic separator and electric dynamo -> magnetically separate manganese from the sluice sand. Alternatively you have to electrolyze tantalite but i think if you can’t find manganese your odds of having platinum to spare are slim.
Edit: I am again retarded, you can find pyrolusite from various sources in a vein pure form or from spessartine/pyrope vein. You can get it from washing numerous ores, tantalite, tungsten ores, and so forth just look for purified pyrolusite ore in NEI. Pyrolusite can be inefficiently processed in a crucible to produce your first manganese.
TL;DR find a dungeon mate it’ll make your life 10 times easier.