If it sounds like a tall order, that's because it is. Dachshunds were bred to go to earth, dig into a den and bark. That's their job. Convincing them that there are other jobs that their time would be far better spent at is the problem.
Of course, like most behavior problems with the dachshund, if they get totally out of control you can just pick them up. Some people take the "Dog Behavior 101" approach, rolling the dog over and growling at it. However, most dachshunds are clever enough creatures to know that this isn't your usual behavior, and it can have the opposite effect if you don't really mean it. Furthermore, you wont' want to do that when you're out and about, so if you correct your dog in all situations, then he or she will learn there are exceptions to the rules.
And that's it. No exceptions. You'll need to let everyone in the house know about it, too. Dachshunds also have a good memory to go with their tenaciousness, making them notoriously difficult to bend to your will. Because of this, punitive punishments are rarely effective as most dachshunds are sure you'll crack eventually. Reward based training is usually the most effective and ultimately, the most satisfying.
Indeed, the key with much of dachshund training is to give them ample reason to do things your way that has nothing to do with making you happy. This breed of dog often has their own best interests in mind, usually food related.
Though many people say their dog is food motivated has not seen the single-minded fervour with which a dachshund pursues food. Were this not the case, there'd likely be far fewer obese dachshunds. You can use this to your advantage when training.
For barking, when your dachshund gets into situations that are sure to provoke a barrage of barking, let your dog know that they'll get a treat if there's no barking. By very calmly telling them no when they do bark, and holding the treat somewhere they can see it, you can reinforce this.
Afterwards, if you see them not barking in a situation, go ahead and give her or him a treat, though it will likely take a great deal of early training to get this sort of pattern ingrained, since barking is a very instinctual response for this breed.
Anti-digging measures are usually best a combination of physical measures to discourage the practice, such as blackberry brambles or plugging holes as well as encouragement to do other things. In fact, a well-tired dog is one that usually feels compelled to less barking. Daily walks can help keep digging to a minimum.
If you have vermin in your yard, it will be nearly impossible to keep your dachshund from going after it - it is their glory. However, unless you want to dig your yard up with a pick axe, control such pests or your dog will. Otherwise, digging is the easiest of the two habits to break, as it simply requires some exercise and attention. Having another dachshund as a play pal will also help let off some of that steam.