Your Trainer Card can receive a number of upgrades, which include new background colours and stars for your card. You have to complete a number of criteria to receive these upgrades. Note that while these are listed in a specific order, you can actually do the requirements in any order and you’ll get a new star each time.
You start off with a basic blue Trainer Card and no stars.
Upgrade 1: Get a record in the Hall of Fame.
Upgrade 2: Complete the Kanto Pokédex, excluding Mew.
Upgrade 3: Complete the National Pokédex, excluding Mew, Ho-Oh, Lugia, Celebi, Jirachi, and Deoxys.
Gold Trainer Card
There is one last upgrade you can get for your card which requires you to play the mini-games located on Two Island. These require the use of the wireless adapters that came bundled with Fire Red/Leaf Green and require 3 players to complete the requirements.
Pokémon Jump: Jump 200 times.
Dodrio Berry Picking: Catch 200 berries.
Trainer Card Stickers
In addition to the stars and background colours, you can also get stickers that go on the back of your Trainer Card where the arrow points. You have to speak to a man in the house east of the PokéMart on Four Island and reach these requirements:
Enter the Hall of Fame: 1 / 40 / 100 / 200 times
Hatch an Egg: 1 / 100 / 200 / 300 times
Win a Link Battle: 1 / 20 / 50 / 100
As an extra, Lorelei’s house (which is also on Four Island) will gain more Pokédolls the more times you beat the Elite Four. You need to beat them 200 times total to get the last Pokédoll, which ties in nicely with the stickers!