I had a holiday that I did not work, but was paid for. I then worked 40 hours in the week, but the employer only paid me at straight time, is this right?

Yes, overtime is only based on hours worked, not on hours paid. Thus, if you received 8 hours of holiday pay, it has not affect on overtime at all. You still need to work 40 hours in the week before you are entitled to overtime pay.

My employer paid me 8 hours of sick time and I then worked 40 hours in the week. My paycheck shows 48 hours of straight time, isn't this illegal?

No. Overtime is only based on the hours you actually work. Sick, vacation, holiday, or any other hours that you are paid for but do not work are not counted for overtime purposes. Thus, it is possible that you legally are paid for 48 hours of straight time if 8 of those hours were non-working hours.