Well, you can cover almost all of your airfare/hotel costs with miles and points from different programs. Very good source of information is a forum named flyertalk.com. Basically airfares are divided into 2 categories: revenue fares ( the ones are sold for cash) and award tickets ( the ones you can pay with the miles). Award ticket inventory is more limited, but with enough advanced notice and some flexibilty in dates and destination you can find the tickets you are looking for. The same goes for hotels, with the difference that it is usually much easier to find a room for points redemption than airfare with award redemption. Every major airline has loyalty program and those programs are affiliated with certain credit card issuer. For example United Continental has several cards issued by Chase, American Airlines has Citi issued cards, Delta is affiliated with Amex, Marriott hotel program has Chase cards etc. Those credit cards offer sign up bonuses. Let's say recently Chase had limited time offer for their United credit card that after you spend $2,000 within 3 months on the card you will receive 55,000 bonus United miles plus you will get 1 mile/ $1 spend, so your total will be at least 57,000 United miles once you meet your minimum spend. The miles get posted to your United account usually in the same or next billing cycle as you minimum spent is met and there is usually no way to take those miles back. First year annual fee is waived, you just close your card 6-9 months after you opened it and this way you avoid paying the fee for the second year. There are also set of different credit cards by Chase,Amex and Citi with proprietorial and transferable points. Basically once you meet the minimum spend the card issuer will give you the number of points in their own currency, which you have to transfer to the airline/hotel partner of that particular currency before you close the card. Otherwise those points will be lost. Chase Ultimate Rewards program has number of good airline/ hotel partners you can transfer to such as British Airways,United Airlines, Hyatt etc.
So, first you have to open up the account with airline/hotel programs and then choose the right credit card for yourself. You will have to learn ins and outs of all major programs to pick the ones right for you as there is no reasonable and cheap way to combine miles/points from different programs. US based airline programs are very good as they usually pass only taxes/goverment fees in additional to miles on award tickets. Most foreign airlines have programs which also pass so called fuel surcharges on "free award tickets" which in many cases make award redemption not a very cost effective option. Generally United and American Airlines will be good choices for your required destination, charging around 60,000 miles/round trip ticket in economy to Europe and around 35,000 to 60,000 round trip ticket in economy to South America. AA also has off-season awards for fall/winter/spring travel which requires only 40,000 round trip to Europe in economy and similar to SA. Taxes on award ticket to Europe are on the high side, depending on destination anywhere between $40 to $220 (UK departure tax is very expensive), SA taxes are a bit cheaper. Every airline program has an award chart indicating redemption requirement for each destination.Since you still have plenty of time before your planned trip you can do some reading and learning, choose the program and right credit cards and start working on accumulating the miles. Chase has limited time offer now for British Airways Visa - 50,000 bonus miles after you spend $2,000 in the first 3 months on your card. BA miles are not very good to Europe, but excellent to SA as you can redeem them to fly on their partners - AA or LAN without fuel surcharges. You can redeem miles from your account for anyone incl. your children free of charge, so basically you need enough miles in your account for 2 people and the same for your wife. She can open her own cards and accumulate enough miles for 2 more people, this way you can take the whole family. And lastly, but important, you do not have to fly the airline whose program you are using to accumulate miles. Those miles are good to book the flights on any partner airline. United miles can be used on Lufthansa, Austrian,Thai,Singapore etc, basically on any airline in Star Alliance and even on some of their partner airlines which do not belong to Star Alliance like Air Lingus