I'm going to ignore the main zone. You can run 7.1 or get a new receiver and run 9.1 or whatever your setup is as appropriate to the receiver. Yes, that's easy.
Zone 2 & Zone 3...
Those are those spaces that people want to be really convenient and sound decent and think they should just work, but they never understand why it completely blows to get audio there.
Specific to the Pioneer Elite VSX-81TXV, it offers one stereo output for zone 2. It is not amplified. It likely has volume control, but that will adjust volume up and down for all connected speakers.
1. Your receiver sometimes, but not often, has amplified audio outputs for additional zones. You should plan on picking up a dedicated amplifier for additional zones.
2. Your receiver may only offer ONE additional zone of audio. So, you need to consider how many zones you have (2), and how to do separate volume control in those zones.
3. Your sources that you play in zone 2/3 must all be connected to the A/V receiver using analog audio connections. This is a HUGE issue with newer devices that only offer HDMI output. When you buy (or rent) make sure you get stuff with analog audio connections so that you can connect them to the receiver and get audio for zone 2/3 output. This is a HUGE failure on the part of HDMI to not require 2-channel stereo along with surround sound.
My recommendation, with what you have... Assuming you don't have volume controllers in the wall for the two zones.
Get a volume controlling speaker selector and a separate amp off of eBay. These parts rarely go 'bad' and can last for decades.
Niles is a great option here, with their SSVC-4, but if looking on eBay, you have to be aware of whether all the phoenix blocks on the back are included. They can be VERY tough to find if they aren't. For example...
https://www.ebay.com/itm/Niles-SSVC-4-Speaker-Selector-Volume-Control/263984052380?hash=item3d76acf89c:g:QtwAAOSwXi9b~aPT:sc:ShippingMethodStandard!20152!US!-1
$65 for the model I linked, but the connectors are gone.
Maybe get the one from Monoprice instead...
https://www.monoprice.com/product?p_id=8232
$65 once again.
This is a GREAT deal! It will handle your two additional stereo speaker zones and provide volume at what may end up being a great price...
https://www.ebay.com/itm/Niles-SSVC-2-Speaker-Selector-Original-Owners-Manual-Included/202582024661?hash=item2f2ad461d5:g:wQMAAOSw7HhcRKuO
$25
Anyway, then you need an amplifier to power those speakers. The Pioneer won't do it.
Literally, any stereo amplifier will pretty much work. Cheap garbage to high end units will do the trick. I tend to stick with major players. Sonance, Nles, SpeakerCraft, etc. You can go with more luxury brands (Emotiva), but they have a premium for what may not be needed.
ie:
https://www.ebay.com/itm/Sonance-275SE-Amplifier/264106188003?epid=117497556&hash=item3d7df49ce3:g:eLMAAOSwtvZcJWZA:rk:18:pf:0
$100.00
This is actually a steal...
https://www.ebay.com/itm/Sonance-SONAMP-1230-2-Channel-Power-Amplifier/302951144607?epid=66773212&hash=item46894b9c9f:g:6bAAAOSw5Ylb5O9A:rk:26:pf:0
It provides 12 channels (up to 6 rooms) of stereo audio, or 6 channels (3 rooms) bridged, of stereo audio. It is one of my favorites to used with distributed audio. While it can power your zone 2 output to the two rooms off of two channels, should you get a receiver with separate zone 2 and zone 3 outputs, it will provide power to those two additional zones separately. It is also designed, from the ground up, for this kind of work.
Anyway, my links will get your speakers powered and give you volume control using what you have for under $200, and there are cheaper ways to go as well if you look around and that's your goal. But, I would always power zone 2/3 speakers from a dedicated amplifier as it takes the burden off the main A/V receiver to help it last longer and allow for more consistent power consumption.
Just make sure you give these components plenty of air to breathe.