You can upgrade your ship in Acarde Mode only, by selecting Arcade Mode then pressing
to access your upgrades.
You gain $1 per point scored in Arcade Mode, so you just need to keep playing Arcade to get more money.
There are 7 seperate areas you can upgrade, some can be upgraded be bought more than once.
Upgrades you can buy:
Health x6
Bullets x3
Bombs x3
Boost x3
Health Pickups x3
Friendship x1
Color Swap x1
Every time you buy an upgrade for an area that can be upgraded, the price goes up each time, so you will just need to keep playing and upgrading, with more upgrades you should be able to last longer and be able to get money faster for future upgrades.
I would recommend personally starting with Health before anything else, but it's all personal preference so you buy what you think is best but also bare in mind the cost.