This is probably going to be one of the geekier posts, which probably only the more hardcore boardgamers will appreciate, and the more normal people will just shake their heads or roll their eyes and think "geek", or "nerd", or "wierdo". These are the things that I usually do when I open a new game.
Punching - I enjoy punching cardboard pieces from the cardboard sheets. I do it with some care, so that I don't tear anything. The pieces must be punched from one specific direction, the direction where you can see the grooves. The other side is the flatter side. You must punch from the "groove-side" because it helps to reduce the risk of tearing. I find that usually games made in Germany have very good cardboard punches. Quite often some of the pieces have already fallen off by the time you open the game box, and the rest are very easy to punch out because they are cut so well.
Sprues (cardboard) - I used to keep them all in a bag, and not do anything with them except occupy space. Then I evolved to keeping them under the box insert. This lifts up the box insert slightly, so that when you close the box, the lid fits very snugly, with minimal space. Your components won't fall off their compartments and get mixed up if you store your games upright (which I do in most cases). Now I give the sprues to my children to play with as stencils. Sometimes I dump them into my recycling bin (for waste paper).
Bagging - I have many zip-lock plastic bags that I bought in Taiwan in 2004, varying from size #2 to size #6 (not sure whether there is an industry standard). I still have not used them up. Most games do not provide bags or do not provide as many bags as I would like, but that's no problem at all. In fact sometimes I prefer using my own bags, so that I can use a standard size. I prefer to use the same sized bags, so that after playing a game, I don't need to worry about which bag to use for which components. If that's hard to achieve, I try to use just two sizes, so that at least it's easy to tell which size for which components. E.g. for Agricola, only one bag size. 1 for all the animals, 1 for resources, grain and vegetables, 1 for round cards + begging cards + major improvements + ref cards, 1 for occupations (2P ones, as I mostly play 2P games), 1 for "used" minor improvements, 1 for "unused" minor improvements (Michelle and I shuffle all minor improvement cards to play, and we make sure we have played all of them before we shuffle all again and start again another cycle), 1 for each player's pieces - people, fences and stables, 1 for all cardboard pieces - food, guests, multiplier markers, 1 for wooden/clay rooms, 1 for stone rooms/ploughed fields.
No rubber bands - One of the basic laws of boardgamegeekdom. Rubber bands melt or disintegrate over time and leave stains on your cards. I bag my cards. I have heard of some elastic bands / elastic tapes which are suitable for cards, but I have never seem them before in Malaysia. Bags work well enough for me.
Box inserts - I usually keep them, but sometimes I throw them away if they are troublesome. I threw away the one for Through the Ages (2nd edition, i.e. FRED 1st edition), because I think it looks nice when you open the game, but it's not helpful when you want to organise the components well. I threw away the one for Race for the Galaxy: Gathering Storm so that I can put both the base game and the expansion in it. A Game of Thrones LCG because it's silly to have an insert which can be used only for your 4 decks of cards, and you have to hide all the other components under the insert. Power Grid because I want to put the France/Italy expansion, Benelux/Central Europe expansion and Power Plant Deck 2 expansion in together. The inserts for Lord of the Rings: Battlefields expansion and Lord of the Rings: the Confrontation Deluxe Edition are poor, but I can live with them.
Catalogues and foreign language rules - I hide them under the box insert. I can't bring myself to throw them away, even the Fantasy Flight Games or Rio Grande Games brochures which I have so many copies of. I have some German version games. Usually I print the English rules or English translation downloaded from the net, and I hide away the German rules that come with the game.
Card sleeves - I usually do not sleeve my cards. Not even for Race for the Galaxy which I have played 200+ games. Actually, especially because I am playing it so many times. I find them a nuisance when playing and when shuffling cards. I can live with some wear and tear, as long as they are uniform. Well, with the Gathering Storm expansion added in now, if I want to I can tell whether it's a new or old card by looking at the card back only, but when I play, I intentionally avoid looking too closely, so that I wouldn't know. I apply the same principle to Carcassonne, which I have a mix of older and newer tiles. I do sleeve cards which will get heavily used - e.g. the role cards in Citadels, the major improvement cards and the round cards in Agricola.
Concise reference sheets - This is something I have been doing since I started getting into the hobby seriously. I do these quick ref sheets which I find handy when I need to teach a game and when I need to look up rules quickly (e.g. how much money you start the game with, or how many cards each player gets at the start of the game). Quite often before I buy a game, I have already researched it and have made a concise ref sheet for it. I'd print it out and put it in the box. I'll do some self-promotion here - the latest set of my concise ref sheets are uploaded to BGG, here.
Shelf space allocation - One of my relatively recently applied policy is that I have assigned one particular section of my boardgame shelves to be the hot section. This section is right at eye level and is easily reachable. Newer games which I expect to play more frequently go here. Some favourites, which I also tend to play more, also go here. I find this system quite convenient. Games which I think I will be playing less will get moved out from this hot section.
Update BGG - I keep my collection on BGG up-to-date.
Maaan... I just love opening new games.
Do you have any peculiar thing that you do when opening a new game? Smell the game? Record the box fart?