How do we move our Sims to other neighborhoods without losing all his/her achievements and skills? Or is it even possible?

You have to save a copy of them to the library, then you can place them in your new neighbourhood.

The sim will lose all relationships with everyone outside of the household, but will keep everything else. Except sometimes they won't a job anymore but other times they keep their job.

And how do I do that? Save a copy in the library, I mean...

You have to do it from Edit Town mode. Once in that view choose Select Tool where you can see the whole town, click on any household and it will give you a number of options on a sub-menu. One is Save To Library. If you choose this option you then have a choice to save the family only, or the family and their whole entire house. If you choose the option to save the whole entire house, make sure that there is a residential lot available in the new neighborhood that would be big enough to accomodate the size of the lot you are copying.

You can see lot sizes by clicking within the box that denotes which lot you are selecting. I don't know how to describe it more clearly--maybe somebody else can or you can learn to do it like I did by just clicking around until the info pops up.

If you move an Author sim without his (or her) house, when you get him into his new house just go to a computer and under "Writing" option, choose the option to let you order books he's already written. They will then be delivered to his mailbox and you can get the books and put them in his bookshelves.

From what I personally have experienced, and I did post asking about this but didn't see any answers last time I checked, the books won't be available as a topic for discussion with the other sims in the new town. LIke, in my old town, I could take any sim up to an author and have them discuss books my author had written. I can't seem to get this option to come up anymore. That's one of the things that gets lost in copying a sim to the new town. He is still an author, he still has access to previously written books, but now they're just not a topic of random conversation anymore...so far as I have experienced.

Oh quick word of advice, if your sim is best friends with another sim in town and you don't want them to lose their relationships, temporarily place the friends into the same household and copy them together into the new town so that they can retain their relationship. You can always copy the friend's house separately and bring that over, too, and move the best friend back into his old house.

If you haven't already, please learn to use the clipboard in the game. It's more versatile than most people first realize and can let you do all kinds of creative saves--like splitting up households into individuals or combinations of individuals for the purpose of saving and then putting the whole household back into the house completely intact.