Within the range of acidity, most vinegars will be reasonably able to substitute for each other, in terms of the foods' chemistry. That means substituting vinegars with similar levels of acid (given as percentages) for each other, or calculating the equivalent amount if the percentages differ. They will differ in terms of the the flavors they give to the dish, but the difference is unlikely to be severe unless the dish is leaning heavily on those flavors, as in, it is a primary flavoring ingredient.
First, there's white vinegar or distilled vinegar - that contains the acidic component of vinegar (acetic acid), but is very neutral in flavor. It can be substituted for any other vinegar for the dish's chemistry... although the percentage of acid is higher, I think, so less will be needed for the same effect. Substituting for this vinegar might be trickier, if the recipe depends on the neutral flavor not competing with other flavors in the dish. Probably white wine or rice vinegar would be closest.
(white) Rice vinegar is described as mild, clean, delicately flavored. It will probably substitute fairly well for white vinegar or white wine vinegar - although it has less acid, and more will be needed for the same effect. The red and black varieties are very different, and I'm not quite sure how they would substitute - perhaps red would be close to apple cider vinegar, while black - no clue, possibly balsamic if its the same as "chinese black" vinegar.
There are a number of, hm, alcoholic named vinegars. Red wine vinegar and white wine vinegar, sherry vinegar, champagne vinegar, and so on. These are generally equivalent to each other - well, they taste different from each other, but a similar range of tastes, I think. They are a little less acidic, and add some fruit flavors to the mix, a little more dry rather than sweet. The alcohol they're made of determines what other flavors get added - not just the broad types (red, white, sherry), but also the quality of wine initially used. Lighter colored (like white) should tend to be milder flavored and more acidic, I think, and darker (like red) should be sweeter and more strongly flavored as a general trend.
Malt vinegar is nutty and toasty, mild and a bit sweet. probably the best substitute would be apple cider vinegar, which is also a little bit sweeter with a fruity taste. By sweeter I mean a little bit sweeter than the wine vinegars, but much less so than balsamic. I would guess red wine vinegar is in roughly the same category of fairly strongly flavored, and a bit sweeter.
Balsamic vinegar has a very distinct taste - it is much sweeter than other vinegars, enough that it is usually substituted with the equivalent amount of other vinegar and a third as much sweetener. It is really not able to substitute for other vinegars unless that extra sweetness can be dealt with somehow. I recently saw a comment that chinese black vinegar might be a reasonable substitute for balsamic, though I have no experience in this.
So, your (white) rice vinegar will probably substitute for white or white wine vinegar just fine, and not need a lot of adjustment - if any at all. The recipes will come out a little different, but not necessarily badly. Your balsamic won't substitute at all, sorry - it's too strongly flavored. You might pick up one of the apple cider/malt/red wine/red rice (and/or black rice?) group of vinegar to stand for something flavored but less sweet, if you have a lot of recipes needing that kind of vinegar - or just use your rice vinegar, and plan to make up the loss of flavor in other ways.