You can get very good sound from good in-wall and in-ceiling speakers, but they carry a premium, and often you put some of these speakers into rooms which are far less than ideal. In-ceiling speakers pointed at hardwood or tile floors in a kitchen with wooden cabinets doesn't always provide much more than 'good' sound, even with decent speakers, but it can all be budget related.
In a 'surround sound' space, where you want good audio, then to get good sound from the in-whatever speakers, you need to spend the money on good speakers. Back boxes and directional speakers can help direct sound and provide a proper backing to deliver the best audio. I'm much more a fan of in-wall vs. in-ceiling because of the directionality of the audio coming from those speakers and that walls tend to be a bit stiffer so you end up with a more controlled surface you are working from.
Anyway, whether you get a separate amp or not is up to you. Marantz, Denon, Yamaha, Pioneer and others sell multi-zone receivers which can handle the THREE zones of audio you are trying to get to. It is important to understand that with three separate zones you really do want a receiver that can handle that. Certainly a pre-amp/amp setup can do this, but many A/V receivers with internal amplification can also handle this and can provide amplification for zone 2 (at least) and audio output for zone 3.
You will want to consider how you are controlling everything and get a remote control solution which is actually easy to use. Your phone is RARELY easy to use in these types of setups. Most zone 2 and zone 3 audio is limited to analog audio connectivity only, so your sources will need analog audio, or a way to feed analog audio to the system.
Wires will need to be run, so plan for drywall repair and painting as appropriate.
Under $5,000 is certainly doable, but it depends on what all is needed. Excellent in-wall speakers can run $3,000/pr. or more, so you know, keep expectations in line with what you really want to get out of the system. Generally 'good' in-wall/ceiling speakers may be $500 each. More 'generic' speakers are around $120/pair or so. You have a lot of room on equipment pricing. If you are shooting for $5,000, then let the installer know. They should be able to fit your budget and to speak to you about what they are getting and why.
Also, let us know your area, and there could be a couple of recommendations here. If you are on Facebook, there may be some community groups which can help steer you towards a good installer as well. Of course, you shouldn't do that if you are in the DC area - just call me.
