Fountain pens
Why write?
Even in the age of computers and Internet access, physically writing with a pen and paper still commands an enormous amount of utility. Firstly, while you may be able to type more quickly than you can write, it does not activate the same centres of the brain, so you may not be able to recall the contents of what you wrote as easily. Secondly, computers are distraction magnets. Sometimes it is helpful to not have to deal with terrible UX, battery life issues or constant notifications. Thirdly, there are certain situations when you may need to take notes, but the use of a computer would be undesirable, e.g. during a meeting at work. Making frequent eye contact with the speaker is important and you cannot do that if you are checking your email. Other people around you may find it distracting as well.
Why use a fountain pen?
While a rollerball or fineliner may offer a passable writing experience, they do not typically glide across the page the way a fountain pen might. This will tire out your arm out more quickly and reduce the legibility of your handwriting. Most rollerballs and fineliners are disposable and thus have a more negative impact on the environment.
TWSBI Eco
I have used many pens from many different brands, but my favourite is the TWSBI Eco, which only costs around £40. It has a huge ink capacity, a smooth JoWo nib and can be fully disassembled. It even comes with a bottle of silicone grease for the piston filling mechanism. It is a demonstrator (transparent) pen, so you can see at a glance how much ink is left. This also allows you to avoid ink starvation, which can occur if you write too quickly for a long period of time; simply hold the pen vertically with the nib facing towards the sky and twist the piston knob slowly to flood the feed with ink. The Eco has a screw-on cap with an O-ring and is quite airtight. This helps to stop the nib from drying out and failing to start when you haven't written for a while.
Expensive fountain pens
When the price of a pen exceeds around £200, gold nibs tend to displace the cheaper steel nibs you might find on less expensive pens like the TWSBI Eco. However, this does not mean that you are guaranteed to have a smoother writing experience. It all depends on the tipping material. Gold nibs tend to flex more than steel nibs and this may improve the look of your handwriting. I do not like this effect and I no longer buy pens with gold nibs. Many expensive fountain pens, especially Montblanc's famous Meisterstuck range, have grip sections made out of cheap, injection-moulded plastic. These may fail over time with heavy or prolonged use. It is important to note that most fountain pens do not retain their value over time, so do not think of them as investments, no matter how pretty they might be.
Inks
There are many different inks you can try. My favourites are Yama-budo and Ku-jaku from Pilot's excellent Iroshizuku line.
Paper
Fountain pen inks tend to feather and bleed through the page more than other types of inks. It is therefore best to pair your fountain pen with high-quality paper that is smooth and less absorbent than regular paper. Note that there is not necessarily a correlation between GSM value and paper quality. Tomoe River is widely recognised as some of the highest quality paper available, yet it typically has a GSM value of only 52. It is also quite expensive, so a more economical choice would be Rhodia, Clairefontaine or Mnemosyne. Avoid Moleskine, it is low quality and unsuitable for fountain pens.
Maintenance
You should clean your pen every few months. This is because ink may dry and clog the feed over time, leading to hard starts or skipping. It is also good practice to clean your pen before changing inks, or to flush out any oils from the manufacturing process if the pen is new. If the pen accepts cartridges or a converter, you should be able to flush the nib and feed out using a bulb syringe and lukewarm, slightly soapy water (use dish soap). Repeat with plain water to rinse off any residual dish soap. Dry the pen thoroughly. Even though the TWSBI Eco does not accept cartridges, it can be disassembled and flushed the same way. For most other pens, you will have to draw soapy water into the feed and expel it using the filling mechanism several times.