I am a recent licensee and I own a Yaesu FT7800 dual band mobile. I want to be able to take it backpacking, but need to know what sort of battery and mobile antenna I should use. Something extremely portable is important.

Thanks in advance.

Good question. Go to the Q&A forum at qrz.com and present this question there. You will learn more than you need to know. Good luck and have fun with your new radio. Welcome to Ham. KI6GRS

A mobile like that is pretty battery-hungry for backpacking. If it's putting out full power from a pack-mounted antenna, you'll need to consider RF safety issues especially on 440.

You can work out all sorts of figures based on the current draw when receiving and transmitting and what percentage of time you'll be transmitting, but in the end, you'll probably find that a 7 amp-hour sealed lead-acid battery makes the most sense. It weighs about 5-6 pounds and you can probably run almost all day on it. It's a very common size for computer UPS's so they're usually the best deal. For a good bit more money, you could go a lot lighter with a NiMH pack that gets 3-4 amp hours. One problem you'll face is that with only 12 volts, most mobile radios are going to have a lower RF output since they are meant to run from 13.8 volts. You can get around that with an 11-cell NiMH pack for 13.2 volts or some companies produce voltage boosters to maintain a level 13.8 volts from a 12-volt battery.

For conserving battery power, a great option is to use a handheld or scanner to receive and leave the mobile radio off until you need to transmit. They draw a lot less power when receiving than most mobiles.

Antenna-wise, there are a lot of options. Any 2m quarter-wave ground plane antenna would work reasonably well on both bands. You could mount one to use the top of a metallic pack frame as the ground plane. A better option is an end-fed half-wave on 2m that won't require ground radials. Something like the Comet SBB-5 is a good choice. It's a half-wave on 2m and stacked 5/8 waves on 440. If you're just planning to operate from stationary points, a roll-up J pole hanging from a tree limb is a nice packable antenna. A portable beam antenna would help a lot, especially if you're just on a handheld radio. Arrow Antennas makes portable beams that pack well, or you can build one even cheaper (Google for cheap yagi).

I have a FT7800 too. I use it with a 12V 7.2Ah lead acid battery and a Diamond NR-770R. The lead acid battery is a gel battery which means that it can be used in every orientation. It can take cycling to the point where the transceiver has to be shut off.
The antenna is a dual band antenna for 2m and 70cm and rated for 100W. It can be taken down to three parts (I usually make it two). With 5W PEP I had QSOs with reports of 57 to 59+ at a distance of up to 30km.
One important lesson was that it´s important to have a sufficient amount of cable to the antenna. A change of antenna placement sometimes yielded 1 or 2 points for the signal strength for both receiving and transmitting (bear in mind that yielding 1 point in signal strength means a factor of 4). Proper antenna placement may help you to save a lot of precious battery power and weight.

I've been looking at similar. Get a small gel cell battery, a small solar charge controller and charge from a small solar panel or feed the output of a power supply into the charge controller when the sun isn't shining.