What would be the recommendations on the more affordable side of the $ spectrum, for the minimum setup to have in your kit? Would you need your own transmitter, or just the receiver?
The answer is situational. At the very minimum you need your own quality ear buds. In many situations, I think you would also want either a preamp/DI or a pedal board fly rig that gives you the ability to dial in your sound.
Whether you need anything else depends on what the band will provide and what you need/want.
A wired solution is generally less hassle, cheaper, and more reliable. IMHO wireless is not really required unless you are expected to roam around the stage. Quality wireless is very expensive and keep in mind you will need a transmitter and receiver for bass and IEM.
Wireless can be problematic. Basically it can be very difficult to get a bunch of mix and match wireless gear to work well together. Ideally you would use the same brand the band uses. It may be necessary to get even further into the weeds. Full compatibility may require using the same series of gear and selecting a unit that operates in specific frequency bands.
As has been mentioned, many digital mixers allow you to control your mix via a phone app. So having a compatible phone and appropriate apps can be beneficial.
In contrast, the band could provide some sort of digital personal mixer. For example Behringer has the PowerPlay personal monitor system. The mixer/headphone amp is the P16HQ
The Rolls PM351 is useful for situations where the band does not provide a discrete IEM mix for each player. It basically serves as a mic splitter, DI, and basic mixer. It gives you the ability to control the level of a monitor mix sent to you from FOH. It also gives you the ability to adjust the level of your vocal mic and bass without impacting the signals sent to FOH. It does not provide any tone controls or protective limiting.
The main advantage the PM351 is it's cheap and gives you pretty good bang for the buck. I would probably opt for a separate mini mixer, mic splitter, and preamp/DI...but of course buying separates will cost a lot more.