Well, firstly, congratulations on writing something by hand and making the recipient's day! Letter-writing is dying out, and I despair for future historians who may have nothing to look at. So much of what we know about social history comes from written accounts - letters, notes, accounts and so on. It will be a real shame if the world loses that ephemera.
Secondly, let's talk a bit more about what you want to write? Because in the same way you don't want to use a spade to prune a rosebush, what you want to write will dictate (to some degree) what you want to write on.
"I dunno. I just thought I'd send a note to a friend to cheer them up."
Excellent! They will love that and it will mean so much more than just a text message.
There are some gorgeous correspondence cards - small enough that you don't need to write a thesis to fill them; beautifully decorated to bring a smile to anyone's face. Even if you only wrote, "Hey there! Just sending you this to tell you I'm thinking about you!" it would be a treasured thing.
There are also some great cards available and I would heartily recommend the Viking sets. Bold, bright, colourful designs that will look great on a shelf; enough space to maybe write a little more; still small enough not to be terrifying to those unaccustomed to writing letters. They're also great for doubling as birthday cards.
"What about if I wanted to write a bit more? I've seen a charity that has people send letters to lonely people/cancer patients and I wondered about doing that."
Again, what a wonderful idea. A simple card would be fine for this too, but if you wanted to write a letter on some beautiful notepaper, I would recommend the Clairefontaine A5 writing pad. The paper is divine, whether you're writing in biro or fountain pen, and it has that thick feeling of quality paper, so the recipient will feel utterly special.
If you're writing in biro, there are paper and envelope sets by G Lalo that are also lovely, but since most of them are laid paper (which has a texture to it) then they don't always play nicely with fountain pen. The exception to that is the G Lalo Velin de France paper (A5), which is satin-smooth and glorious.
"Okay. I've really got this letter-writing bug and I want to write a long letter. What do you have for me?"
Well, there's a selection of A4 pads and envelopes. As I said, it depends on what you're writing with (biro/fountain pen) to some extent, as textured paper really can be a pain in the proverbial to write on with a fountain pen.
These are textured papers:
G. Lalo Toile Imperiale (linen-texture)
G. Lalo Verge de France (another laid paper)
Bomo Art Letter Set (laid paper; beautifully decorated envelopes)
These are silky smooth:
Whatever you choose, the person you are sending it to will no doubt treasure it. I still have letters sent to me decades ago, and when mail arrives and there's a handwritten letter in there, it's always a very special day.