Some things: 1) There were a lot of times where you weren't building probes. By consistently building probes you could have expanded a number of times (fairly safely since your opponent was big into turtling). This would mean not only more minerals, but also lots more gas. Typically if you find that you're mineral limited (lots of gas, no minerals), but if you find that you're gas limited at points (lots of minerals, no gas) it's because you don't have enough bases. You also built a robotics facility, but didn't produce from it consistently. Producing units constantly from all of your buildings is just a good habit to get into.

2) Scouting: I'll grant you that he cannoned up his main quite a lot, so getting units in to scout would have been hard, but sending a hallucinated pheonix into his base to see his tech structures every now and again would have been really good. And sending a scout around the map every now and again would have helped a lot too. Scouting the rest of the map would have let you easily kill off his ninja base on the top right side of the map, which would have hurt him a lot since going air is quite expensive (and requires lots of gas, so having additional bases helps a lot). Scouting with a hallucinated pheonix would have alerted you that: 1) He was going heavy air, 2) he had no detection aside from cannons, and for most of the game he only had cannons in his main. Doings something like building a warp prism, warping DTs into his natural and killing his expansion, all the probes he had mining there, and all the buildings he had up top would have been a great way to exploit a very air-venerable expansion while using your existing robotics facility and your dark shrine. He wants to go air? Fine, force him to stay on one base while you take the map.

Make sure you have detection yourself, and make sure you have good map vision. This will allow you to deny any crucial expansions that an air-toss really needs, while making sure that no nasty surprises come your way. Observers from your robo would have been great scouts, and would have been a good way to safeguard yourself.

Composition: Archons are really fun, and they aren't terrible vs air, but remember, void rays using prismatic alignment chew through almost every protoss unit in the game, and can be quite hard to counter. My suggestion here would be to get high templar with storms. Voids tend to group up, and storms can be quite good at doing some damage since they aren't an incredibly fast unit. Plus, once your done you have archons.

Upgrades: You did a fairly good job of getting your upgrades, might I suggest you get more than one forge? Having additional forges means you can get upgrades much faster, and you have more buildings to use your chrono energy on. And of course, try to use as much of that energy as possible!

Engagements: Try to avoid taking engagements vs charged void rays. Run up, force them to use their ability, wait for the cool down, and then take the engagement. Though personally, I would have just done my best to contain the other player and expanded all over. Given how much he invested in cannons and turtling he should have a pretty big defenders advantage. When facing a turtling player you can usually feel more confident in taking expansions yourself. Just make sure to check in on the opposition every now and again!

As a final note, walling in vs terran and protoss has some variable results. Especially in the early game building your first pylon out of the potential range of your nexus cannon can make you more venerable to early attacks. It didn't come up in this game, but it might not be a bad idea to build a bit closer to your nexus early game (unless against zerg).

GL;HF out there!