Don't prune anything just yet or at least not the ones that have flower buds on them. You have quite a few of them already. If you decide to prune, only the branches that don't have any flower buds on it. But, if you mange to bundle them up, just do that and prune the tip only after bundling them. Looks to be a mature plant already, with that many flower buds. Where are you located, FL, CA or some other warm state? You should have ripe fruits in a couple of months.

Any other tropical plant/ fruit trees, came with the new house? And if you do cut/prune, don't discard the cuttings. You can root them easily by sticking them in the ground. But also offer them up for trading, shipping fees, etc. There are many people that would love to trade/ buy or pay shipping fee for a cutting or two from you. If you decide to prune for trading/selling or shipping charge, cut the long branches into 8-12" long cuttings. They don't only have to be tip cuttings, one very long branches can be cut up into many pieces of the mention size, to fit the box. BTW, have you gotten any ripe fruits from that mature plant and know what the inside flesh color is, red/pink/white/magenta? And was the previous owner Asian?

its looking a bit yellowish
I would throw a teaspoon of epsom salt on the soil near the roots
maybe some fish fert, or cow manure

here is a link with some info on fertilizers + other stuff...
http://edis.ifas.ufl.edu/hs303

this site says low nitrogen since its bearing fruit,
but since it hasnt had any ferts and its yellowish,
you could probably give it some.

For bearing trees
It is important to apply low amount of nitrogen and high amount of potash for obtain ample yield
http://www.agridept.gov.lk/index.php/en/crop-recommendations/1077

Here is a link that might be useful: dragon fruit plant

Thank you Green and Soaht!

Will look into the fertilizer.

I guess my main question would be what is the best way to manage these? Bundling them right now is just not possible, there are too many thick limbs and we're afraid bending them would damage them.

Should we wait until after fruiting to start cutting away all the limbs?

Our plan was to cut away all the smaller weaker limbs without flowers. Do we cut close to the base?

Also, when cutting the tips, do we cut just above the joint? Or cut below the joint?

To Soaht's question... we're told they're red flesh, and yes, in southern california.

Red flesh ones are usually sweeter than most white flesh ones So you'll like them a lot better. If you only ever had store bought bland white flesh ones and didn't like them. Well, your mind will change, when you guys get to try a home grown ripe fruit.

Let them be as of now, let the plant finishing fruiting and ripening them. Then you can hack away the plant of any limbs, you don't want. Those smaller thinner weaker limbs, are actually new growth, that hasn't quite mature and thus is why they are weak looking. If you can bundle or tie them to the trellis and make them climb up and over the trellis, they will naturally and eventually stick on to the wood post with their aerial roots.

You can cut some of those none fruiting thick branches off. Cut close to the main stem/branch, that is going up the wood post. The new growth will be easier to train to climb up the wood post/trellis later.

As for cutting the branches, you can cut above or below the joint. They will root either way, but before you attempt to root, let the branches/cuttings heal/dry and the wounds have closed for 3-5 days. This will prevent the newly wounded exposed part from rotting, once you can put them in the ground.

And if you do offer them up for trading/buying/postage charges, you should wait until people actually want them, so then you can make fresh cut and the cuttings will be fresher won't be drying up too much.