There's a good web site for information about portable oxygen at <a target=_blank class=ftalternatingbarlinklarge href="http://www.portableoxygen.org/">http://www.portableoxygen.org/</a> .
The issues with the portable oxygen concentrators are:
1. The batteries are expensive and don't last very long (not even an hour). So if you don't want to be plugged into electricity (car or home), you need to buy lots of batteries and replace them over and over again.
2. Only some of them are approved by the FAA for flying and what's approved is not consistent among the airlines
3. They weigh too much to be carried. They come on wheels so you drag them like a suitcase.
4. As far as flying with them and nothing else, you need enough battery power for the duration + 1-2 hours (runway time).
Short article just on the 2 most popular oxygen concentrators <a target=_blank class=ftalternatingbarlinklarge href="http://www.portableoxygen.org/july.html#lifestyle">http://www.portableoxygen.org/july.html#lifestyle</a> .
As an aside, the lightest portable units are not concentrators, but the smallest compressed O2 tanks. You cannot use them on an airplane. You can have a milkcarton full in your car .
I think your choice should be whatever makes it easiest for you to do your day to day activities outside of the house. It depends a lot on when you are using o2 (24/7 or just at night - but remember just at night eventaully will become 24/7), how you travel most of the time (if you only fly occassionally it might be easier just to rent for that occassion), whether your electricity goes out a lot, etc.
Don't forget that these days you are renting to buy because you will end up owning whatever unit you rent and your insurance company will no longer pay for rental of the equipment then (but it will still pay for repair of equipment and the O2 refills if you haven't chosen a concentrator).