Alright, I think I have the general picture. Before I offer advice on how to reduce biting, I like to get an idea of what the bite triggers are and the general history of the bird so I can hopefully work out the source of the problem. Parrots definitely do bite, but they almost always have a reason. In many cases, biting is defensive in nature - it is a way for the parrot to protect themselves or something they value (like their cage, their food, or their mate) from a perceived threat. But there are many different things that can lead to biting, and the proper approach depends on the reason behind the behavior.
Cage aggression is pretty common among parrots, because they will often view the cage as their personal space and protect it from any intruders. I suspect that the biting isn't purely territorial in your case. It is also related to trust level. From what you've said so far, it sounds like your parrot is still relatively new to you and doesn't fully trust that you are safe. When your hands come into his cage (his territory), he probably feels threatened and wants your hands to leave. Since the bird can't fly away from your hands, his choices on how to respond are very limited. The biting is a reaction to that stressful situation. Because putting your hands into the cage can reliably trigger biting behavior, it can be considered a "bite trigger".
Ideally, you want avoid bite triggers completely and work on developing trust, so the parrot becomes increasingly more comfortable with your hands and normal handling. Sometimes it isn't possible to completely avoid a bite trigger - for example, you must reach into the cage to refresh food and water each day. And you don't want to do that while being repeatedly attacked. Not only is that not good for your hands, but it is also going to reinforce the undesirable pattern of behavior and could lead to escalating aggression as the bird gets frustrated by your repeated intrusions. There are various creative approaches to dealing with aggression inside the cage. For example, if your parrot attacks when you reach for his food bowl, you can try using a distraction. Set-up a treat bowl in a far corner of the cage. When it is time to change the food, place a desirable treat in the bowl and wait until the parrot is investigating the prize before retrieving the food bowl. If you time things right, he will never even notice that you were in the cage. And if not, at least he is farther away from your hands, so you have time to get out safely.
For cleaning the cage and other things that require more time inside the cage with a bird that is protective of his territory, the best approach is usually to allow your bird out-of-cage time and do cage maintenance when he is not in residence. The cage should be his safe spot and where he goes for food and sleep, but it shouldn't be his whole existence. Even untamed birds can be offered time outside the cage, although you will want to be sure to bird-proof the entire room and have a plan for how he will get back into the cage afterwards. You don't want to force him back inside, because chasing, toweling or grabbing the bird will scare him and increase his fear of your hands. Ideally, you want him to choose to return to the cage on his own. Most birds will go back to the cage eventually, because they are hungry or tired. Letting the bird out in the late afternoon or early evening, when you have no other obligations and nothing else that needs to be done for the rest of the night will allow you to wait for him to explore his freedom and hopefully figure out how to return to the cage at his own pace.
Continue to spend time with your bird and watch his body language closely to see how he responds to different interactions - is he relaxed and happy or upset and scared or something else entirely? Try to find things that he enjoys and identify things that frighten him. Do your best to make your interactions positive and avoid pushing his boundaries too much. If you find yourself getting nipped .. you are probably pushing too fast. Back off and give the bird more space. In fact, I would recommend against touching the bird's wings just yet. From what you've described, I think he may be tolerating the touching right now, but it doesn't sound like he trusts your hands enough to allow that kind of handling for long. I would be careful. You might get bitten if you continue. With a nervous nippy bird, it is better to wait for him to invite your touch, rather than trying to see what you can get away with.
From the sound of things, your relationship is still pretty new. I think that if you are patient and build trust, you will see steady improvement. The main thing is to try to learn from the biting and avoid repeating actions that trigger defensive behaviors. Move slowly and talk to your bird in a gentle reassuring voice. Explain what you are doing and stop if the parrot's body language is telling you he is getting ready to attack. Your goal should be to learn how to prevent bites by anticipating your parrot's behavior and reading his body language. Not only does this stop you from getting hurt, but it will help your parrot to trust you and communicate his needs to you without biting.
Good luck and I hope things get better with your new conure. They are such pretty birds. Do you happen to have any pictures to share?