My favorite tip to pass on to new parents is to invest in 4-5 waterproof bassinet- or cradle-sized mattress pads. I first started using these at home and keep one at every diaper-changing station in the house (one upstairs, one downstairs). They provide more coverage than regular changing pads and since they are waterproof, they work better than blankets. I can use them on the floor, the bed, or anywhere and not worry about leaks. Cleaning is easy, just throw them in the wash periodically and use a little stain treatment if needed. They are also pretty cheap to buy and last a long time. I have been using the same ones for a year now.
Later I swapped one out with the so-called "changing pad" that came with my diaper bag. The waterproof pad is thinner and folds up more compactly, taking up less space in the bag. But since it is larger overall it does a much better job of covering up the public restroom changing table or floor. This is especially handy when your baby gets bigger and wriggly and those itty-bitty changing pads cover their backs and rear ends only.
Using these will also let you create your own "diaper" bag if you don't like what you see out there that's labeled a "diaper" bag. I got a larger women's bag from a discount store that was cheaper than either of the two "official" diaper bags I had at home. It holds everything I need for the kids and me and fits my style too.