But there are some bikes with all up and some with all down, right?

Answer 1

Short answer? No. Nearly every bike you will ever encounter has first gear down (from neutral) and the remaining gears up.

Longer answer? Yes, but they're rare, or specialized applications.
Racing bikes often have a reversed shift pattern. If you're turning left, and leaned way over, you'd rather have your foot above the shift lever, about to push down (to shift up) than below the shift lever.

Some older bikes (my first bike, for example, a 1974 Yamaha TY80 trials bike, was neutral on the bottom, and four gears all up) have altered shift patterns for various reasons. Many older British bikes shifted on the right, and were one up, four down. But shifting was standardized in the 70s. There are very few exceptions to the one down-the rest up rule.

Answer 2

Some bikes, pre '70s had all up transmissions. Newer 50cc bikes with semi automatic trani (no clutch lever, but you still shift gears) have one way shift patterns.
Bikes changed to a 1 down 3/4/5 up, because when down shifting fast, you'd end up in neutral on the older style bikes. Not safe for emergency, quick acceleration.

Answer 3

There used to be some bikes where you pushed down to shift up the gears.

The reason was that roadracing bikes worked that way because you might want to shift up while leaned over in a corner, and your foot might hit the ground if it had to move under the lever. You were less likely to want to downshift while leaned over.

But it was so confusing to have two different ways to shift that they passed a law and now all street bikes shift up to go up the gears.

Answer 4

with the exception of the early FS1E (fizzy) all bikes are one down, and rest up. I take it you've not passed your test?