How can I drill a straight hole without a drill press?
If you are set on using a hand-drill, there are jigs available that will keep the drill perpendicular to the object being drilled.  Alternatively, holding a level against the drill will help you keep it aligned.

I would suggest using a Forstner bit instead of a spade bit.  Spade bits can jump around a bit, tear out and are not that precise. They are good for rough work, but a Forstner bit will give you a much cleaner cut.

That being said, a drill press is the best tool for this, and they can be purchased relatively inexpensively.  You could also find a  "maker" workshop in your area, or a college with a shop where often you can rent time and use their tools.