What are some good ways to get rare materials?

Apart from what Lasse said:

Strange coins:
Doing Nightfall and Weekly Heroic Strike is a most efficient way to get those. Weekly on max difficulty is a guaranteed 9 coins at once. Nightrall is a bit more random, but has a chance to give coins, exotics and old ascendant materials (and a weekly exp bonus!). See more on wiki.

Ascendant materials (old ones):
There are plenty of those in Vault of Glass raid (and some good loot too). If you can find a team that knows what to do, then it doesn't take much time to run it. It has 4 loot drops and 3 chests (4 if you can beat Templar without teleporting). So you can end up with plenty of materials. See more on wiki. If you don't have a group, then try Destiny LFG, there was an official Bungie blog post about this site, it helped lots of people to find a team.

Also, if you're short on ascendants but have some spare legendaries or exotics, then don't forget to level them up completely (no need to purchase perks) - legendary guns have a chance to give up to 6 Energy on dismantle (up to 3 if vanilla), and any such exotics will give 2 shards (only 1 if vanilla).

Vanguard Marks
Vanguard marks are best farmed from strike playlist. The best one right now is Roc - it gives rep, marks and guaranteed engram. If you do those, not only you get your 100 weekly marks quick, but you also get lots of engrams (basically you increase your change of getting legendary gear through sheer number of engrams you get, and cryptarch packages when it levels up). If you don't have The Dark Below, then Tiger playlist is your best bet, it has less rewards though, but same amount of marks.

This is one of those places you should be choosing your battles. Unfortunately it isn't really an option to fully upgrade (using ascendant) every legendary and exotic you have or come across.

So make sure that the items you do fully upgrade are the important ones. Test them out and make sure they have modifiers that are worthy of the materials.

Once you have identified the worthy, here is some general strategy.

• Do the daily heroic every day. It will yield 2 ascendant materials and these add up quickly.
• Don't be a hoarder! Dismantling less than desirable legendaries yield ascendant materials. Weapons will yield 2-3 energies, and Armor will yield 2-3 shards. This excludes legendaries that are obtainable by glimmer alone. If you have time, level your weapons prior to dismantling them. Do not spend material on them, but gain experience on them so that all skills are unlocked. This will double the amount of materials yielded (5-6 energies for legendary, 2 exotic shards for exotic), and if you fully unlock (without actually purchasing) all the circles you will also receive a telemetry for that type of weapon.
• Do the weekly heroic because it will give you 9 coins. Do the weekly nightfall as well, because it essentially guarantees a legendary, exotic, or sometimes up to 11 ascendant materials.
• You can solo the first chest of the raid!! What? No. yes. Load into the raid, make it to the loading circle that spawns the bridge. Cross the bridge. Enter the darkness crazyness. When you load into the darkness you will see a path and a light beacon looking thing. While aimed at that, go in the direction of 10 o'clock (as in to the left a decent amount) and run straight until you get the wall. There will be a door that opens to a closet sized space that can have a chest in there. If it is not there, die and try again. It shouldn't take more than 5 tries. Only good once per week.
• So, you did all of this? Good job Guardian! :) Now make another character than can also farm the material by getting carried by a friend so you don't have to invest in them. This will double or triple your weekly income of materials.
• Motes should be easy to come by, and in the process of all of the above you should have leveled to generate them or found a few at the cryptarch. Really, motes are only good for gambling with Xur (and even then, not terribly useful after you are all gambled out).

The new Ghost system provides a new way to supplement your materials. Every legendary Ghost will have a material column in it's skill tree. You can choose to either

Detect nearby spinmetal\helium filaments\spirit bloom\relic iron\wormspore

or

Gain a material when looting engrams in the cosmodrone\moon\venus\mars\dreadnaught

Also worth noting that Ghosts also provide a Glimmer boon for killing specific enemy types, so if you can get a ghost that provides Spirit Bloom materials from engrams and glimmer from Vex, you can use that during Venus strikes and get more glimmer and spirit bloom without doing any extra work. I'd recommend getting a ghost with each material ( and hope for a related enemy type) that you can switch between for strikes and it will help you hunt your materials.