So what is the best way to do that? Do you unlock characters as you naturally play through the missions? I don't really have a great understanding of the game. When playing online, I see people who have impressive Warframes and gear, whilst I seem to be stuck with the weak defaults! I'm enjoying the game more though; I've figured out how to use upgrades properly which reduces the 'underpowered' feeling.
In order to unlock new Warframes, you must obtain several items. These are:
Warframe Blueprint
Warframe Chassis
Warframe Systems
Warframe Helmet
Orokin Cell
The Warframe blueprint can be obtained from the store and varies in price per warframe from 25000 to 35000 credits.
The chassis, systems and helmets all drop from specific bosses. Which bosses drop which Warframe component blueprints can be pretty easily found through Google searching.
Orokin Cells are found in certain later game missions and are quite rare, or can be picked up through certain alert missions.
Additonally, you need the materials and credits to make the individual components. These vary greatly, but generally you'll probably need to grind certain missions for some of the rarer ones and have quite a large amount off credits to hand, at the very least.
I think that just about covers how to unlock Warframes.
In order to unlock weapons, you must once again buy the blueprint from the store for a varying amount of credits, but weapons are usually made up of the simpler materials that Warframe components also use and so are easier to obtain. Note also that if something in the store has "dual" or "ak" before its name, you'll need to make two of the items and then combine them both, for example, "Dual Heat Swords" are made by making two individual Heat Swords out of their components then buying the Dual Heat Swords blueprint to combine them. Generally, what I'm saying is that you shouldn't waste credits buying a "Dual" or "Ak" blueprints until you already have 2 copies of the individual weapon version.