Is it cheaper to book two one-way tickets?

Rarely. Stick to a single 'multi-city' ticket on one airline or alliance to protect your booking if one leg is canceled.

When is the best day to book?

Mid-week, specifically Tuesday or Wednesday, often sees lower prices, but consistency in tracking is more important than the day of the week.

Should I use a travel agent?

Only if you have a very complex itinerary with multiple connections. Otherwise, self-booking gives you more flexibility to make changes online.

Booking flights for a couple's trip to Europe

Book your flights 3 to 6 months in advance using a 'multi-city' search tool to avoid backtracking. Always book directly through the airline's website rather than a third-party aggregator to ensure you keep control of your reservation.

Use the multi-city search tool. Don't book round-trip tickets to the same city. Select 'multi-city' on search engines like Google Flights. Fly into your first destination and out of your last to save time and money on return trains.
Monitor prices with alerts. Set up a tracking alert on Google Flights for your specific route at least 4 months before your trip. When the price drops to a baseline you're comfortable with, book immediately.
Check individual airline hubs. If you are flying from the US, look at the major airline alliances (Star Alliance, Oneworld, SkyTeam). Use a search engine to find the route, but cross-reference the price on the airline's own site before clicking purchase.
Confirm seat assignments. As a couple, you want to sit together. Verify that your booking class allows for seat selection. If you book a 'Basic Economy' fare, you likely won't be able to pick seats until check-in, which often separates couples.

Is it cheaper to book two one-way tickets?: Rarely. Stick to a single 'multi-city' ticket on one airline or alliance to protect your booking if one leg is canceled.
When is the best day to book?: Mid-week, specifically Tuesday or Wednesday, often sees lower prices, but consistency in tracking is more important than the day of the week.
Should I use a travel agent?: Only if you have a very complex itinerary with multiple connections. Otherwise, self-booking gives you more flexibility to make changes online.