Round the world tickets & multi stop flights
Not all round the world tickets can be bought online, but some can these days.
Check the prices of RTW tickets on these websites:
What types of round the world tickets are available?
There are only a few types of RTW flights available.
The most common are the BA/Qantas tickets and the Star Alliance fare.
Or, consider upgrading to business class round the world flights.
Does it matter which way around the world I fly?
No. If you buy a round the world ticket travelling westbound (flying to America first) then it would be no different than travelling eastbound (flying to Africa or Asia first).
If you choose a west-bound RTW ticket then you will wake up much earlier in the morning with jet lag (and get tired in the afternoon). Whereas if you fly to Asia first then you will struggle to get up in the morning and stay awake much later.
Planning Multi-stop flight tickets
Not all gap year backpacking trips or sabbaticals need to go around the world. Quite often a multi-stop flight is all that is required and this can be a much cheaper option.