There may be some cities that still have this, but I think the popularity of the "Welcome Wagon" died down after the internet made information so available. Used to be that when you moved into a new house the "Welcome Wagon" would arrive with informational pamphlets about city services, school info, coupons for local businesses, and maybe some key chains donated by a car dealership. Basically, they would bring you information about the community.
In a previous ward, I was in the Relief Society Presidency. We were very close to a chiropractic college, so several times a year we had a large number of young families move in at the same time. In the spirit of the "Welcome Wagon", we would put together folders for new move-ins. It wasn't anything extravagant or expensive. We would copy a (fairly recent) ward phone directory, a copy of the most recent Relief Society monthly newsletter (that the secretary put together), a list of important contact phone numbers in the ward (Bishop, RS President, Missionaries, etc), a local (free) family magazine, information about the nearest temple (directions, hours, etc), and a list of local attractions (zoo, parks, museum, etc) & their hours/cost, etc. We put everything in a plain colored Pee-Chee style folder (with pockets inside) and attached a label (printed with something like, "Welcome to the ______ Ward!" Members of the Presidency would present a folder to each new Sister when they made "welcome to the ward" visits to their homes.
We made a dozen or so of these ahead of time so that we could grab a few as we needed them.
This idea could easily be adapted for a "welcome" folder for newly baptized members. You may want to coordinate with your Bishopric, RS Presidency, or EQ Presidency to organize something like this. Or you (with approval from your leaders) may want to take the project on yourself. It would be great to have folders for both new move-ins and some for newly baptized members! Newly baptized members have probably already been in the area for some time, so the contents of their folders would be different (they wouldn't need a list of local attractions, for example).
Here are some ideas of what to put inside folders for newly baptized members:
Ward list, list of important ward/stake contact phone numbers, a Gospel Principles Book (student manual for Gospel Essentials class), recent copy of Relief Society newsletter (for Sisters), current handouts/upcoming ward/stake events, pass-along cards to give to friends, maybe a welcome letter?
If you'd like to put together a "Welcome Wagon" folder to give to new move-ins and/or newly baptized members, here are a few additional suggestions:
-I would usually buy a large quantity of the colored pee-chee style folders during the early back-to-school sales (usually July). They come in basic colors (red, blue, yellow, green) and are usually around 10 cents each during the sales...they are a LOT more if you buy them any other time of year! Try to make a guess-timate of how many you will need for the year. Buying enough for the whole year when they are cheap will help to keep your costs down.
-A lot of wards/stakes don't use annual printed Stake directories anymore because the cost is high, and by time that they are printed much of the information is already out-dated, anyhow! You can easily print a current ward directory from your ward website (accessible to all church members through www.lds.org). If you print the whole directory as-is, you're going to have a lot of pages to print & copy. There is an option to print an abbreviated version, which is really all you need for a phone directory...its the same thing, except that it lists only the adults, so its a lot more compact!). You will probably will want to update your master copy several times a year, since phone numbers change frequently these days!
-Finally, I should probably mention that since you're a representative of the Church, items included in your folder should not be commercial in nature (unlike the original "Welcome Wagon"!) For example, it probably wouldn't be appropriate to include advertisements for a business (such as an AVON catalog or a coupon for $10 off pest removal services), even if its a business that you (or another ward member) is associated with. This isn't meant to be a sale pitch, and it should not be used that way. Even if its meant innocently, it will come across as inappropriate (because, frankly, it is). In our Relief Society folders, we included information about local zoos, etc, because although those are businesses, we felt that in our particular circumstances, our suggestions were generally informational in nature, instead of being sales-pitches. In general, it would probably be best to get the final approval over the contents of your "welcome" folder from your local leaders, just to be on the safe side.