I would personally get a bike first, as there isn't a super large need to have more than 3 rows of inventory space. With bikes, it will save you a lot of time when travelling from one area to another and can be super useful. Backpacks on the other hand are useful, but not a priority as you can always sell items to free up your inventory space. However, if you have access to fly (if you're a champ+ rank), then obviously bikes aren't as important and backpacks would come first.
I highly recommend you to get a bag instead as you can have more space for carry on items which is a good thing Collect and save up lots of stones and potions, great and ultraballs as well, but basically it is my personal opinion, the main choice is based on your own opinion. If you are not a champion rank or above, I would suggest getting a bike as it's cheaper and transportation would be much easier, if you are a champion rank or above, it would be a waste of tokens to get a bike as you can fly
I personally would get a bag, as you can store more pokeballs, potions, etc. Bikes are not the easiest to get around on if you want to go slow, however city-to-city wise, they are much faster. My suggestion is that if you're not Champ rank or higher, get a bike as you can always sell items to get more inventory space, and if you are Champ or higher, get a bag as you can fly.