Do you have to have previous tiers?
Do you have to upgrade the tiers for multitool, exosuit, or ship upgrades in order? For instance, will tier 3 still have an effect without the previous tiers (tier 1 + tier 2). I ask because I keep finding stuff that has only the higher tiers and I don't know if the tier has an effect.
It would save me a lot of slots if I could just use the highest tier without the lower ones.
Only one tier of an upgrade is needed to unlock the capability on your multitool/exosuit. So yes, using a tier 3 upgrade will work just fine, even if you do not have the tier 1 or 2 upgrades.
Different tiers do stack, however, which is why you often find tools with multiples of the same augment. For example, if you have tier 1 and 2 sprinting stamina upgrades on your exosuit, you will be able to run longer than if you only had tier 2.
As others have pointed out, you do not need the previous versions. The decision about whether to stack upgrades is actually a bit nuanced:
Suit hazard protection: Say you apply multiple heat deflector upgrades. Sure, you get extra protection from heat, but now you have two separate items to monitor and keep charged up. If you go with only the stronger upgrade, it will give you less charge, but you'll only have to monitor that one item.
Ship shields: Your ship's shields, on the other hand, have only one item that needs recharging (the base shield item), no matter how many upgrades you apply to it. You don't have to monitor the upgrades independently. (In my experience my ship shields take a huge amount of damage from enemy ships, so the more upgrades the better.)
Suit shields: Same argument as with ship shields, except that in the case of your suit you can't manually recharge your shields even if you want to, so there's really nothing to "monitor". (In my experience, my suit shields never take that much damage, so a single upgrade was fine.)
Life support: Even in supposedly "harsh" environments, a single upgrade has been fine for me.
Jetpack: Apply them all!!! :)