Comfortable, dry shoes are an absolute necessity these days. Lose your mobility, and you're just counting time until death inevitably catches up with you. Whenever I run across shoes, I check them for size, and try to keep a good spare pair on me at all times.
Your choice of footwear is a bunch of trade/offs - protection vs weight. Sturdy boots are excellent in combat or when moving in rocky or debris-laden terrain, but over long distances their extra weight can meaningfully add to fatigue. Running/tennis shoes are excellent for speed and agility, but they wear out quickly and soak up water like sponges.
You'll hear people advocating for barefoot-running occasionally. If you've already trained this way and it's working for you, fine. But this is not something to experiment with when hospitals and antibiotics are in short supply. You *will* fuck up, you will probably get an infection, and you may die.
Bikes have a lot of things going for them. They can increase your speed and range significantly - even a modestly-geared bike can allow a fit survivor to outpace any Infected. They're easy to jump off and abandon if you need maximum mobility or for combat. They're typically light enough that you can carry them past inconvenient terrain or debris.
If you can get a sturdy mountain-bike, you can do a decent amount of off-roading, though be careful - they work best for one thing - going *down hills/mountains*. They can actually waste your energy if you ride over mushy ground/thick grass/up hill. Plus, remember why you're out. Scavenging should be high on that list, and you generally find more stuff where there are roads anyway.