What are an amateur's options to put a high-end IC on a board?
What's your budget? You can always breadboard using breakout boards as mentioned in another answer, but you'd maybe get half of those 24 ADC bits as useful information that way. Getting even 16 useful bits out of an ADC requires a PCB with good power and signal integrity practices.

You could try your hand at laying out your PCB, but it might quickly get overwhelming.

Alternatively, TI offers an evaluation kit for the part which will have been designed to squeeze as many bits as their Application Engineers could manage. See here:

ADS1256 Performance Demonstration Kit

From reading the User's Guide on that page, it looks like the kit comes in two parts, which gives you some options. The smaller board is the ADC and its supporting circuitry. I've only skimmed the guide, but it looks like the analog signals are broken out on J1 and the digital signals on J2. Thus, J2 can be breadboarded to your Arduino (it's a 100mil pitch, which matches your typical breadboard, so you might just have to turn the board upside down and plug in), and J1 can potentially be wired directly to whatever you're measuring or preferably broken out on a board to a proper cable connector. The digital control is just I2C so your Arduino will have no trouble.

It looks like you used to be able to buy just the smaller board, but it's been obsoleted. You can get both boards together for $200 from TI. The second board is a DSP-powered interface to your computer. You'll have to read up on what that offers, but you might even find that's all you need for your purposes, assuming it offers an SDK and you're not trying to do any remote monitoring. If you're cost-sensitive and just want the smaller board, try contacting TI directly. They are often accommodating with demo kits, especially if you're associated with a big-name institution or company.



Lastly, since I work for a TI competitor that rhymes with "shmanalog shmevices" (nowhere near their chip business though), I'll just add that when looking at evaluation boards it's worth checking out as many devices as you can. You might find some eval boards more useable than others for your applications, depending on how the Apps Engineer wanted the demo platform to be used.