Start with a Parker
There are hundreds of fountain pens. For a first pen, a Parker is the right answer for most people. The nibs are consistent, the cartridge system is standard, converters (for bottled ink) are widely available and the price is honest. You do not need to spend more than $40-60 for a pen that writes every day for years.
Cartridge or bottled ink
Cartridges are small ink-filled capsules that click into the pen. They are convenient, clean and a good way to start. When you are ready to explore more colours, a converter lets you fill the pen from a bottle. Many Parker pens accept both - you can start with cartridges and move to bottled ink when the colour bug bites.
What to expect the first time
Uncap the pen, press the nib to paper at a shallow angle (around 45 degrees) and write. It should flow immediately. If it starts then stops, the ink is not flowing yet - try writing a few small circles until it starts. If a pen has been sitting dry for weeks, flush the nib with water and reload.
Maintaining a fountain pen
Rinse the nib and converter with cold water every few weeks or whenever you change ink colour. Never use hot water or soap. Store the pen cap-on and ideally nib-up or horizontal. A fountain pen that is capped properly can sit for weeks without drying out.