When you start your game, the rucksack will have 15 empty slots available. The remaining 15 slots will be unavailable until you go through the steps to upgrade your rucksack.
You can upgrade your rucksack as soon as you obtain the Animal Whistle and meet the Harvest Goddess at her pond. If you go back to Harmonica Town and visit Flute Tailors, you'll find Yolanda and Shelly in front of the store. They want to help you out, so Shelly volunteers to work on upgrading your rucksack if you bring her the required materials.
Shelly would like you to bring her Wool and 10 Green Herbs so she can sew a better rucksack.
The quality of sheep's wool you bring her does not matter. You can either get it from a sheep on your own farm or buy the Wool from Horn Ranch. The cheapest Wool that Hannah sells is Decent Wool for 1240 G.
The Green Herbs are a wild item that randomly appears each day. Some days you may find Green Herbs, other days you may not. You have to look around the typical 'wild item' spots until you've collected 10 of them.
After you collect the materials, bring them to Shelly at her store and give them to her. She will increase the size of your rucksack by 5, to a total of 20 slots.
The next day, return to Shelly's shop and talk to her again. You can activate the next rucksack upgrade event without having to wait for any other requirements or events.
This time, Shelly will ask you to fetch her some more difficult items; 1 Perfect Wool, 5 Bodigizer, and a Honey.
The wool, again, can come from your sheep. Perfect Wool is never for sale at Horn Ranch, but if you are in your first Spring season you can buy Perfect Wool at the Animal Festival on Spring 28. The wool costs 2440 G and it is the only time the Perfect Wool is for sale at a shop.
Bodigizer can be obtained a few ways. The easiest is to buy it from Choral Clinic for 430 G each. The most difficult way is to make the Bodigizer using your kitchen. The recipe requires a Red Herb, Pontata Root, and Honey mixed together using the Mixer cooking utensil. Pontata root is another uncommon drop from the Moles inside the mine or you can buy it at the Fall 27 Crop Festival.
Shelly requires any quality of Honey. You can buy Decent Honey from Marimba Farm (not from Horn Ranch) for 100 G each. The other way is to grow a lot of flowers on your farm, and randomly in the morning you may discover bees floating above them. If you pick the flower you'll get honey instead.
Bring all the items back to Shelly and she will expand your rucksack to its largest size. You will have all 30 slots available to use.