The oyster card sounds complicated, but once you figure out what you need, it is easy to use, convenient and will save you money. You get a full refund, including the deposit you pay for the card, but there's a caveat. 1 If you need to "top up" the card, you should always pay the same way that you did originally (either cash or the same credit card). 2 Make sure you don't have more than £10 left on the card when you're ready to return it. You can return it at just about any tube station. You will get a refund on the spot (the balance plus the deposit) either cash or a credit to your credit card, depending on which you used to buy it. If your balance is over £10, or you mixed payment methods, they'll mail you a check in pounds, and your bank will probably charge you more than what the check is worth to process it. If you think you'll be back in the next couple of years, you can hang on to the card for next time. BTW if you aren't sure, you can ask the bus driver or tube agent to read the card and tell you the balance.
Which oyster to get and how much to put on it? Don't buy the visitor card. It costs a lot to have it shipped to you. Buy a regular card at a tube station in London. You can buy it at Heathrow when you land and use it right away to take the tube into the city. You want "plain vanilla" - not contactless, no Oyster account. This page has pretty clear explanations. If you are in London for 6 days or more, a 7-day travel card saves you money too. Here's a chart for fares. Most tourists only travel within Zones 1-2, so that's usually the best buy. The ride from Heathrow is more zones, the card will figure it out. The Oyster will automatically calculate the lowest fares. If you are going to be in London for more than 7 days, when you get the oyster you can ask that the 7-day travel begin on a specific day, not necessarily when you buy it. For instance, if you won't be doing much traveling on your first day, and going on a day trip out of town the next day, you can buy the 7-days when you get the oyster and have it start on your Day 3. Take a deep breath, go through it slowly. Once you realize your options are limited, you can ignore the tons of irrelevant information. Then you just hop on a bus or take the tube whenever and wherever, no worries, no calculations.
The navigo is a weekly pass, beginning on Monday. Do not get the visite pass designed for tourists. You'll need a small photo (about 1"x1.25" - I used a photocopier to reduce a passport photo) to insert in the card when you buy it. You can buy it at most metro stations (but only from an agent and often the stations are not manned, especially the smaller ones). This UNofficial site seems to have reliable, up-to-date info on the navigo and is easy to follow, will answer all your questions.
If your 9 days in Paris are mid-week to mid-week, it's possible that the navigo won't save you money. But it may be worth the convenience. While Paris is more 'walkable' than London, there are still considerable distances and your feet can get pretty tired too. With the navigo, you can hop on a bus for a couple of stops, or just ride it for a "free bus tour" of different parts of the city, without thinking whether it's worth the price of a ride. You don't have to fumble for the individual tickets or mix up the new and used ones.
Other cities - each one is different. In Florence you'll easily walk everywhere. In Rome, a travel pass can be useful. In Venice, a vaporetto pass nearly always saves you money. Where else are you headed?
