No, "think about" something means to have it in mind or consider it. What do you think about the government? I am thinking about Jean (whom I might like, or dislike, or be neutral about). I think about food often!
To think well about something means to consider it thoroughly. Think about your decision well, because if you don't you could make a serious mistake! I thought about it well, and I've decided not to go after all.
But be careful--to think well OF something means you approve of it, which is quite different. I think well of Ryu, who asks intelligent questions, for example. I don't think think well of certain other people, however (brings to mind my first example of thinking about the government)!
At the risk of sounding too technical, I'd like to know if "think about .." involves "giving a good thought to .." or it is just a combination of "think" and "about."
Of course, the act of "thinking" itself always involves some will to make a good use of your brain, but if "think about" is just a combination of "think" and "about," it would not seem to involve a "good thinking." In Japanese, "think about" or to be more specific its translation into Japanese, is simply a combination of "think" and "about." "Think" in Japanese ("kangaeru") does not necessarily imply good thinking.
You said it is "consider," and consider is "think about something carefully" according to Longman.
In conclusion, "think about something" is "do a good thinking" + about "something." What do you make of it?
(Perhaps, some misunderstanding was involved. I used "think well" by meaning "think good.")
I think I am going on a trip = I have the intention of going on a trip, I will probably go on a trip.
I am thinking about my trip. (think + about, really just a combination of the two words; it implies nothing about the quality of one's thought or approval or disapproval of the object of one's thought). The "about" let's you distinguish this from the previous example.
I am thinking well about my trip ==I am thinking about it quite a bit, or quite carefully. "Well" here means carefully, or thoroughly.
I think well of the idea = I think it's a good idea. "Well" here indicates your approval of the idea.
I hope this does it? I think part of the confusion is with the words "well" and "good", which can mean so many different things, depending on context.
And now, I am thinking well about going to bed (I am carefully considering the idea), because it is almost midnight and I need to sleep.
Now I have decided! In fact, I think well of the idea (I think it is a good idea, I should do it)!