Hero learning order?

Maybe just go through the motions of doing the tutorial completing the limited hero pool matches first (but perhaps random in those). They are good heroes for someone new/just coming back to the game.

Start with ranged supports with hard CC.

Generally they don't need much farm, everyone likes having them on their team, nobody really cares when they inevitably feed and for most of the game they are by far the most important characters on the map (map control is the #1 way ensure victory and vision is the first step to that). They usually have the following jobs:

• buy and upgrade courier
• buy wards and place them on the map in areas that will be important for the next few minutes
• lock someone down so your team can kill them
• stun someone so your carry can escape impending doom
• provide a distraction so that your carry or person on godlike streak can escape
• provide gank support (chain the stuns, etc) to secure advantages in each of the lanes

They may also have these additional jobs:

• stack jungle camps for people to farm
• stack/pull the easy camp to control the safe lane equilibrium
• build arcanes/mek/force staff/hex to use at appropriate times

Good choices for this: Crystal Maiden, Jakiro, Vengeful Spirit, Lina, Lion, Shadow Shaman

If one such support is already going to be picked and you have some synergy elsewhere, a different kind of support is welcome (either no/soft cc like Lich, Keeper of the Light or Venomancer or melee like Omniknight, Treant Protector or Ogre Magi). I will not discuss them further because this book of a post would be too large.

Eventually you will cross paths with others who understand supports are the real backbone of the game and you will get teams who have already picked 2 before you get to do so.

Don't pick a third support.

I mean, yeah it works sometimes. Normally though, a team with 3 people not looking for some active farm gets crushed. No matter how good Crystal Maiden, Jakiro and Lina are separetly or as pairs, a game with all 3 of them together (all playing the "we don't need last hits" game) will feed the Sniper /Viper /Drow Ranger /Phantom Lancer /Lifestealer /Phantom Assassin on the other team more than you can handle.

In this case, the next most important heroes on the map are the ones who can control teamfights. Sometimes they play mid, sometimes they play offlane. Most of them have a soft CC spell or some kind of positioning spell. It just so happens that most of the heroes who aren't hard carries or hard supports fall into this area (and some of those others do as well). Unfortunatly that means there is a very wide skill range to the heroes here as well.

Nevertheless there are a few who I would recommend getting to learn here: Clockwerk, Dark Seer, Magnus, Dragon Knight, Batrider, Viper, Axe, Queen of Pain, Puck

A lot of players like these heroes because they tend to be the ones that have the highest KDA. While it might be cool to see a mega kill announcement or a godlike streak for your hero, understand that it is because of a bad play on the opponent's side or good play on the part of the supports that got you there. A Viper can do nothing if he is way behind because he was ganked a couple times early or has no farm due to being out of position and/or having poor vision allowing him to see incoming flanks.

A high KDA or GPM or XPM does not win the game for your team.

The only thing that wins the game is taking down the enemy ancient. I'd say this almost always happens one of the following ways (which may overlap):

1. Supports magnify some advantage (of these there are 2 that are regularly used: superior vision in a particular area of the map, the only variable that the players can have direct control over, and the capability of the opponent to survive a direct assault in or near a creep wave) into a team wide gold/xp/tower lead to snowball to victory.
2. One team pushes for an objective and in doing so leaves open an exploitable weakness that the other team takes advantage of (Have you ever base raced a Nature's Prophet or Anti-Mage? You will.).
3. Due to poor play, one player on the opposing team gets too big for your team to handle. 

The role of carry

Imagine dota was a game with a time bomb on both sides. Eventually someone gets big enough that nothing can be done to stop them from taking chunks out of your base. For the first 40 minutes of the game, this person appeared to do nothing at all. Half the time they weren't visible on the map, the rest of it they were off in some other lane tapping creeps on the shoulder. They have a KDA of like 1/2/2 and you have pretty much ignored them most of the game because they just weren't doing anything threatening (warning: that is a lie) or dangerous. Suddenly they appear on top of you, hit with a nuke and one or two auto attacks and you are dead. Then they pop a bkb and eat the cc from the rest of your team and drop to about 1/4th hp. Then they activate satanic and are at full health and your team is dead.

That is a carry. Their goal in the game is to reach this end state before the one on your team does and before losing. Some of them participate with their team more than others, some arrive after teamfights to pick up straggling "leftovers".

Well, that is the dream anyway. Often the game ends before the carry's dream comes true. As a carry it is your job asap or win the game before it happens. It turns out that lane farm is the fastest way to do it (aside from feeding). This is why being able to last hit is so very important for a carry.

Here are a few carries worth learning to pick up as a new player: Gyrocopter, Luna, Alchemist, Lifestealer, Faceless Void

Those 5 tend to be with their team more than some of the others (Anti-Mage, Spectre).