The most traditional way to prevent sediment from winding up in our glasses is to decant, standing the bottle up for several days before opening, and then pouring the wine slowly and carefully into a decanter in order to leave the sediment behind. A viable option to that, especially with very mature wines is to also stand the bottle up for several days (that giving the sediment a chance to settle to the bottom) and then pouring with great care directly into the glasses.
It is not at all uncommon in France and in Italy to pour wines containing sediment through a cheesecloth. Metal filters can also be used but this is frowned on as materials such as tin and lead may well impart a flavor to the wine and, even though stainless steel will not, the probability is that household filters will not do the job very well. More common however in the USA is the use of cheesecloth and paper coffee filters.
Whether to decant or not is a somewhat rambunctious subject - some in favor of decanting nearly all wines to let them open, some even in favor of double-decanting (first into a decanter and then back into the bottle) as that adds even more air to them. Others (including myself) see little reason to decant wines at all, that with the possible exception of very young wines and decanting then is done quite differently - simply by turning the bottle upside down and letting the wine pour as rapidly as possible into the decanter, that giving it a "blow of air".
As to differences between young and mature wines - the major one is that the more mature a wine is the less it needs air and decanting in some cases can be quite harmful, even causing the wine to lose its charm within minutes.
My bias: If your hand is steady enough to pour into a decanter, it is steady enough to pour directly into glasses. And oh yes, although some will disagree, never, never decant a white wine from Burgundy.