Many plumbers, electricians and carpenters pull out a chisel and cut a notch. It's fast, and you don't need to use power tools. Use a wide chisel, at least 1-inch wide or wider. Mark the pipe's location on the joist to start. Draw a rectangular box that's slightly bigger than the pipes on both sides of the joist with the opening at the bottom. Use a hammer and chisel to score around the drawing on both sides and at the bottom to a depth of about 1/4 inch. Use the hammer and chisel to cut the drawing out deeper on both sides and at the bottom. Continue cutting around the opening until it's about halfway through. Tap upward from the bottom with the chisel and pry out sections of the opening, splitting the joist along the grain between the cuts on both sides until the notch is complete. Use a file or back saw to trim and clean the opening if necessary.
Select the location for the notch on the joist. Insert a spade bit or hole saw into a drill/driver that's at least as big as the diameter of the pipe. Drill through the joist with the hole saw. It's OK if you leave some material on the joist along the bottom. If you don't have a bit or hole saw that's wide enough to accommodate the pipe, drill two or more holes side by side. Use a chisel and hammer, back saw or hand saw to cut the hole square or rectangular. Cut up from the bottom to square off the sides of the notch. If the top of the notch has a semi-circle shape or is irregular, use a course file or small saw to flatten or square it.
Reciprocating saws are power saws that look like a saw-nose shark. They have a long blade extending from the front that cuts through almost anything. They're easy to handle. You can cut curves or squares almost effortlessly. Draw the notch where needed on the joist. Cut up from the bottom of the joist using the blade of the reciprocating saw. Cut up both sides to the top of the notch drawing. Cut up from the center of the drawing next, but angle or curve the blade into the corners of the notch. When the pieces on the sides of the notch drop out, turn the saw horizontally and cut across the top of the notch. It might take two or more cuts from both directions, but the reciprocating saw cuts fast and smooth for notching joists.
Use hand saws for convenience. Draw the notch on the joist. Use a back saw, which has a short, stiff blade for best results and ease of handling, or an ordinary hand saw; either works fine. Cut up from the bottom first, to the top of the notch. Hit the joist hard with a hammer between the cuts to pop the notch out. Sometimes, the piece will break off at the top leaving a flat square notch. If it breaks off below the top, use a hammer and chisel to chip the remaining wood out of the notch.