First-time traveler - what plane ticket should I get?

Which one is better depends on what is important to you so nobody else can tell you. However there are a few things to think about that might help you decide. The big difference between multiple individual tickets and a return (or routed) ticket is who is responsible for breaks in the journey. A few years ago my daughter came back from Australia to Ireland for Christmas. Bad weather meant her return flight from Dublin to London was delayed until after her onward flight from London to Australia had taken off. Because she had bought separate tickets, she was considered as a no-show and had to arrange her own alternatives. Had she bought a through ticket, the airline would have been responsible for getting her home. Whether that matters to you depends on you and your journey.

This might seem obvious, but planes only fly from airport to airport, so think about how you intend to get to and from the airport at both ends. That will often help you decide which airport you actually want to fly to. RaynAir are renowned for getting you close but not necessarily close enough to your destination. Stanstead to London will cost you a few hours and quite a few quid. As will Girona to Barcelona, Beauvais to Paris, or Charleroi to Brussels. If you know what to expect before hand and have planned for it, then it isn't a problem, but if you didn't know, it could turn into a nasty surprise.

Each time you have to change on a journey adds stress to the journey, so while you can often save a little money taking the round-about route, only you can decide if the extra stress is worth the savings. When I was younger, I used to love taking weird and wonderful routes because they often brought me to exciting places I had never seen before, now-days I prefer fly as near direct as possible even if it means changing the dates I travel.

How flexible are you? The more rigid your plans the less choice you will have and (usually) the higher price you will have to pay. Ask yourself if you really need to be in a given place at the same time everybody else wants to be there. Late May on the Cote d'Azure is magical, unfortunately you have to share it with the Cannes film festival so everywhere is overpriced and booked out. The first week of June is just as magical but without the crowds.

Are you happy to organise every thing for yourself, or are you willing (and able) to pay for somebody else (a travel agent) to look after all the details for you. For your first major journey, I would recommend a travel agent. They can do so much more then just book your flights, including most of the things you haven't thought of yet. I would suggest you do a little preparation first. Know where you want to go and when. Know what you would like to do when you get there (tickets booked in advance are usually much cheaper). And know what is important to you and what you can be flexible with. Listen to their advice but make your own decisions. Finally, have fun.

When comparing two or more plane tickets, you use such criteria as price, number of stops (layovers/transits), safety history, customer service reputation, on time performance of the airline, airport flying out of and how close it is to your house and destination, quality of their frequent flyer program, penalty for rebooking and re-pricing, brand name, etc.

Typically you want lower price, fewer stops, good customer service, low penalty for rebooking, generous frequent flyer program, excellent safety, etc. You get the idea.

Everyone will have their own mental utility function giving different weights to the various criteria depending on importance to them, just like when deciding which brand of car to purchase. The rational goal will be aim at the combination that gives you the highest satisfaction U (utility).

U=f(x1, x2,...xn)

I for example will never fly Aeroflot even if I was offered first class tickets for $10 because the anecdotal horror stories I have heard about them (likely not true) from my youth have mentally traumatized me.

do we use the same plane ticket to get home?

Yes you usually use the same plane ticket to your destination and back. Probably 80% (it is falling with the advent of budget airlines and budget fares) of tickets purchased are return/round-trip tickets.

Why it’s sometimes better to book two one-way airplane tickets instead of one round-trip flight

The percentage of one-way tickets purchased has risen steadily since 2014, and a report by the New York Times found that 44% of airline travelers booked a one-way ticket instead of a round-trip ticket

Return tickets are usually cheaper than two one way tickets and sometimes even cheaper than a single one way ticket. The reasons for that will be left out at this time.