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Your base's buildings and building levels are key to unlocking the best and strongest units. To be able to upgrade more buildings at a time, save up Machine Spirit (MS) to get a second and third techmarine.
The general strategy for upgrades is to prioritize upgrades that make your attacking force stronger. This makes it possible to attack all the buildings on your map, losing fewer troops. That way you can do more attacks (and thus earn more glory) in any given amount of time. The second priority is to make your defense stronger. A weak defense makes you vulnerable to others attacking you, which makes you lose glory and resources.
But it's important to prioritize attacks first, since most players you attack will not attack back, so being able to destroy more bases more easily is a better path to higher glory.
Following are the priorities you should generally follow when deciding what to build/upgrade. Note that certain circumstances may change the relative priorities of these so consider these guidelines.
Always upgrade your HQ as soon as possible. This will unlock all the other upgrades below.
Always upgrade your research building to the highest level possible, and upgrade the troops you use most. You should always be researching something. Once you've upgraded your favorite troops, upgrade the combat tactics you use the most. While not the top priority, upgrading mines can also help defend your base (more on defense later). It's a good idea to research other troops too, since that will give you the flexibility to use those troops if needed. Once you get to higher levels, units that haven't been upgraded are too weak to be useful, so you need to upgrade your troops before starting to use them.
Having all your RZ upgraded to the maximum level should be top priority. This ensures you get the most units possible for your level, making your army stronger for assaults and skirmish. Because you can't recruit new units on upgrading RZ, your attacking army is weaker once you lose the units on an upgrading RZ. For this reason, it's tempting to skip RZ upgrades, but this only makes your army weaker in the long run.
Tip: Before upgrading a RZ, put it on a separate callsign (preferably the purple triangle since it deploys last by default) and recruit units that survive well. Then you can choose to deploy those units only if needed, and after all your other units have made it safer for them to deploy. If things go poorly, don't deploy these units but withdraw instead.
Upgrading the warlord throne makes your heroes better. Heroes are a key part of successful assaults and skirmish.
Upgrading the logistics building gives you access to more of the map, making it possible to get more glory more quickly.
Tip: after upgrading your logistics building, you will have many new areas to scout, at the cost of munitions. While your research building is upgrading, you are not spending munitions on upgrades, so this is often a good times to scout new territories. Note that to do this you may want to upgrade logistics before research, despite it being a lower priority.
Upgrading the forge sometimes gives you more relic pedestals. These allow you to get more passive bonuses to your troops and buildings. Upgrading the forge also gives you more wargear and relic slots so you can have better choices when assigning them.
Upgrading the scriptorum makes your defending heroes stronger, and gives you more hero slots.
While this is generally low priority, be careful not to neglect these upgrades. The Command HQ typically costs a lot of power, and this increases with each level. Keep your power storage upgraded so you always have enough power for the upgrades that become available. Also, when upgrading rally zones, it can be hard to collect more power if your army is weaker. So if your power storage is only enough to do one rally zone you will need to rebuild your whole power supply with a weaker army.
See Defending Your Base for details.
Note: Upgrading turrets do not fire. But they can still act as a distraction for the enemy attackers units.
The vault protects power and munitions from being captured by enemy attackers. Key if you are attacked a lot.
At early levels, you will rely on these to get power and munitions. But as you progress you can get more and more power/munitions from base assaults, and upgrading these becomes low priority.
Tip: Avoid upgrading too many of the same type of building at the same time. For example, if you upgrade too many RZ, your attacking force will be weak, and you won't be able to get a lot of glory. Similarly, if you upgrade too many turrets at once, your defense will be significantly weaker. A good rule of thumb is to upgrade one RZ, one turret and one other building (e.g. power storage, forge, etc.) at the same time (assuming you have 3 techmarines).