Warframe has a lot of different resources you need to farm in order to build new Weapons and Warframes etc. Some are easy to acquire just by running your ordinary missions, other are more rare and require you to focus your effort.
I highly recommend that you use Nekros (or better yet: Nekros Prime) whenever you are farming for resources, as his Desecrate ability will give you a second chance to loot resources from the corpses of your enemies. Also make sure to equip weapons with high Slash damage as this will cut off body parts and the Desecrate ability will loot every single body part.
If you don't have Nekros yet, you can use Hydroid with the Pilfering Swarm augment mod instead; Hydroid is very easy to get for new players - if you don't have Syndicate standing to buy the Pilfering Swarm augment you can buy it for Platinum from other players. Use Hydroids #4 ability and kill enemies trapped by the tentacles in order to gain more loot.
This rare resource only drops from the Orokin Void tileset and from certain Assassination targets, e.g. Jordas Golem on Erris (make sure to break all containers near boss area).
I recommend you use Nekros and equip weapons with high Slash damage as this will increase your chance on getting Argon Crystals. I prefer to run either the Exterminate or Capture missions in The Void when farming Argon Crystals - make sure to explore the entire map and break all containers.
Important: Argon Crystals decay over time - after 24 hours half of your Argon Crystals will be gone! - so only farm them when you have all the other materials required for a blueprint.
Each mission will give you a small amount of credits. However, the fastest way to get credits is running The Index on Neptune. I recommend you choose the Medium Risk Investment when running The Index as this seems to give the best result of credits compared to the time invested (High Risk Investment give you more credits but also take longer to finish and has a much higher risk of you failing). When the Void Trader Baro Ki'Teer visits a Relay Station, there are many players that need lots of credits to buy rare mods from him, so these weekends are the best time to run The Index, as you should have no trouble finding a group.
You can buy blueprints for Detonite Injector in the Chem Lab of your Clan Dojo and build them in your Forge on your Orbiter ship.
You can also get a fully build Detonite Injector from Invasion missions (you need to run the same mission 3 times and then the loot will be mailed to you when the invasion is over).
You can also get Detonite Injector as reward when doing bounties in Orb Vallis.
Rare and Reinforced Grineer Storage Containers will usually also contain a Detonite Injector but as the name imply they are very rare.
This is rare resource drops when defeating Lech Kril (War, Mars) or Tyl Regor (Titania, Uranus).
You might also get it from other missions on Mars or Uranus; I recommend trying the survival mission on Uranus - bring a Nekros and stay for as long as you can while keeping the Desecrate ability up all time.
Mutagen Mass are available as bounty rewards when doing bounties in Orb Vallis.
You can also get Mutagen Mass as a mission reward when doing Infested Invasions - check which missions reward Mutagen Mass (it is a rare reward so only a few Infested invasions will reward it!) and then simply run the invasion mission 3 times - the loot will then be mailed to you when the invasion is over.
You can also buy blueprints for Mutagen Mass in the Bio Lab of your Clan Dojo and build them in your Forge on your Orbiter ship. However, in order to build them you need 10 Mutagen Samples, which are also a rare resource (see below) and you might need those Mutagen Samples for unlocking more research in your clan Dojo so I can't really recommend building Mutagen Mass from blueprints until your clan have completed all the Bio research.
Mutagen Sample can be farmed from Derelict missions. I recommend the Orokin Derelict Survival mission; in order to gain access to this mission you must first buy the Orokin Derelict Survival Blueprint from the market and then build some in your Foundry.
The easy way to get Nitain Extract is just by doing the daily and weekly Nightwave challenges and then buy it from their offerings: 5 Nitain Extract cost 15 Nightwave creds.
If you manage to find all three hidden Resource Caches on a Reactor Sabotage mission you might get lucky and get some Nitain Extract.
During a Ghoul invasion on Plains of Eidolon you can run Ghoul Purge bounties for a chance of some Nitain Extract.
You can't actually farm Platinum but you can farm rare mods and Prime parts, which you can then sell to other players for Platinum. You can buy rare mods from the Void Trader Baro Ki'Teer or augment mods from Syndicates, which you can then sell to other players for Platinum. You can also farm Void Relics and then run Void Fissure missions to unlock Prime blueprints - keep the ones you need to build the weapons and warframes you need and sell duplicates for Platinum.
I recommend you use warframe.market to list the mods and Prime blueprints you are selling as this is much better than using the in game trade chat.
This is a very rare resource. The best drop location is Ophelia on Uranus. This is a Survival mission - try to stay for 30-40 minutes as it seems drop chances improve the longer you stay.
I recommend you use Nekros when farming Tellurium, as his Desecrate ability gives you a chance of extra loot.