Water pump to intake size

I really appreciate everyones help, don't what I do without this site, so thank you to everyone.
The piece I need is the bypass nipple. What I have are 3 different sizes.
1. The water pump appears to be 3/4 when I pulled the hose off it was rigged up with a smaller house cut off, and the new house put right over top of it to accomodate the smaller size on the water pump.
2. The Intake bypass nipple appears to be 1" as the normal hose slips right on it.
3. The heater core hoses are the smallest and say 5/8 right on the hose.
So what I need is a hose to connect the water pump to the intake as well as a bypass nipple all of which to accomodate the smaller size that the water pump is. Also can I use any 3/4 hose?