The addition of up or down in these types of phrases can be either literal or figurative, depending on context.
By literal, I mean that the person is literally ascending or descending in space. Usually this means ascending or descending a hill, ladder, staircase, elevator, etc. For example, if I'm on the 3rd floor of a building and Frank is on the 2nd, I might text him saying, "Come up to the meeting room, we're waiting for you."
There are several figurative uses of "go up/down":
- To indicate cardinal directions: up indicates north while down indicates south. Up stands for north because of the traditional orientation of maps showing north toward the top of the map. Someone who is traveling from Los Angeles to San Francisco and back might be said to "go up to San Francisco", and to "come back down from San Francisco." This is usually used when speaking about longer distances of travel, especially between different cities/states.
- To indicate travel towards or from a city or hub of a city. In British English (possibly specific to London in particular) one is said to go "up to town" or simply "up town." In American English, the word downtown stands for the central business district of a city, and is often used as in "I'm going downtown this afternoon." (See here for more information about downtown.)
- To move toward a person's residence. Colloquially, at least in my American English experience, up and down are often used interchangeably here regardless of cardinal direction or elevation. Example: "We're going up to Marilyn's house."
Since you mentioned that you draw your examples from Oscar Wilde's well known play The Importance of Being Earnest, I'll examine those examples in that context.
Garden at the Manor House. A flight of grey stone steps leads up to the house. The garden, an old-fashioned one, full of roses. Time of year, July. Basket chairs, and a table covered with books, are set under a large yew-tree. (source)
CECILY. [Smiling.] But I see dear Dr. Chasuble coming up through the garden. (source)
Possibly, Dr. Chasuble is literally coming up, in the sense of ascending a hill. In context, Cecily is probably closer to the house than Dr. Chasuble, and Wilde could have written it this way to say he is coming "up" toward the house.
CECILY. Well, I know, of course, how important it is not to keep a business engagement, if one wants to retain any sense of the beauty of life, but still I think you had better wait till Uncle Jack arrives. I know he wants to speak to you about your emigrating.
ALGERNON. About my what?
CECILY. Your emigrating. He has gone up to buy your outfit. (source)
In the previous act, Jack was in London, where he remains at the time of this scene. Cecily refers to him having gone to London as having gone up to the city from their country house in the British style I mentioned earlier.