I'm getting my first Mac, a Macbook Air 2015. My other machine is a PC used for gaming and I thought I could use my external HDD with both. Now I see NTFS is read only in OS X. For now it's unlikely that I'll run out of space on the 128GB SSD because I stream both videos and music and don't save large collections of photos or videos on the notebook but if might need some more space, what would be the best way to upgrade storage? Move the files on the HDD to my PC, format it to another file system and only use it with the Macbook? Does OS X have a utility for that? I heard it's possible to upgrade the SSD but will it void my warranty to have it done?
Your best option is to get a memory expansion module. I have a 2013 Air and I purchased the PNY 128gb StorEDGE. It's very fast and would double the available storage on your Air.
As others have indicated, opening the Air to add another/bigger SSD would be more expensive and difficult than just adding an expansion module.
You can reformat your HDD in FAT or exFAT format (from the Mac or from the PC), so you'll be able to use it with both machines. BTW the software to use on the mac is Disk Utility, in Applications /Utilities