Training pay will vary wildly from company to company, but will certainly be lower than what you will make as a solo driver.
Be prepared to travel very lightly while in company training, as space on your trainers truck will be extremely limited for trainees.
It will vary by company, length of training, and the type of job you're doing. Below are a couple of examples.
Swift Company-Sponsored CDL Training Pay
Trainees earn a guaranteed weekly salary of $600 per week or 12cpm (combined miles between trainee and instructor), which ever is greater.
Trucking Company Reviews - Celadon Training Pay
You will average approximately $450 per week during your training. Trainee pay is based on 10,000 miles at $.18 per mile.
You're going to want to bring as little as possible, as you will be on the trainers truck and most of the space will already be taken up by his stuff. So one good-sized duffle bag, pillow, & blanket is most times all you're going to be able to bring.
Wiki - Items To Bring To CDL School, Training, and OTR
Lack of storage and living space is probably a driver's biggest consideration when packing for CDL school, on-the-road training, and solo life on the road. Drivers will generally pack lightly for school and training, and their company should give them a general list.
Forum - Tips and tricks for cab organization
When you get on your trainers truck you will find all sorts of things. I've seen microwaves bungee under the top bunk... passenger seats taken out for cabinet space.