When going on a RTW trip, you will be crossing many borders as you will be visiting many countries. Border crossings can be stressful. If you don't have the required documentation, you can be stopped from boarding the plane or even worse, arrive and be denied entry to the country, resulting in a trip back at your own expense. It is obviously very important to check border entry regulations for each of the countries you will be visiting and making sure you have a valid passport and the appropriate visas and documentation to enter. Check with the consulates of the countries you will be travelling to and double check as the date of travel nears. Border regulations are constantly changing so it is important that you have the most up-to-date information.
It often helps if you dress nicely, cover up a bit and avoid clothing that other cultures might find offensive. You should also, always be polite and respectful to border officials. It, also, is a good idea to make sure everyone in your group is on the same page in terms of your onward travel plans, even if they haven't been fully decided yet. Better to have a cohesive story that might change, than to have each of you telling border officials different stories and arousing suspicion. Also, be sure to have any required documentation on hand in your carry-on and not packed away in your checked luggage.
Visas usually require applications to be submitted before you depart, some of which can be quite extensive. Be thorough in your research regarding visas and if you're unsure of what you're doing, it can be helpful to hire an agency to assist with these. Note that it is your responsibility to obtain the appropriate visas.
Many countries, such as Peru, the UK, the USA and the Phillipines, among others, also require proof of onward travel. They want to see that you actually intend to leave their country within the allotted time period. So what if you've booked open or flexible tickets or are planning on booking a train out of the country in a few months when you know the exact date you want to leave? What if you don't have a ticket for travel out of the country? Proof of onward travel typically doesn't require you to have a ticket out of the country in question. Simple documentation that you intend to continue travelling and will not be staying in the country should be adequate. A booking for a flight three months later out of a neighbouring country back home should do. These rules are in place to stop illegal immigrants from entering and working in the country, not to cause issues for tourists, though it obviously can affect you if you don't have the required documents. Always double check to make sure, in case the country in question requires something specific or in the event that they've changed their rules regarding what is accepted.