Differences in train prices

I was looking online for the prices in train, for example, from Asakusa to Shibuya, but it display different prices. Like for one, it would display 460 yen, while another one would display 510 yen, and another one 390 yen. Why is that? Is it because it has different routes? Moreover, when I am at the train station, how do I specify which one I want? I am trying to go to Japan with a budget, so I need to spend as little as transportation as possible. Thank you

Why is that? Is it because it has different routes?

Yes. Because of different routes and different train companies used. Basically: the fewer companies you use, the cheaper it will be.

Moreover, when I am at the train station, how do I specify which one I want?

If you use just one company, just purchase one ticket that gets you to the destination station:
http://www.japan-guide.com/e/e2016.html

If you use multiple companies, you have to buy separate tickets for each company when transferring. However, sometimes, it is possible to buy just one "transfer ticket" for multiple companies, for example, if you transfer from Toei subway to Tokyo Metro.

If that all sounds too complicated, you should consider a prepaid card, which is valid on all companies:
http://www.japan-guide.com/e/e2359_002.html

Thanks, that was really helpful. Another question I have though, is about the train's time. Are they always the same, everyday? Or do they put out new schedule every week or so?

The schedules do not change very often. Only when a new line is being opened or extended there will be amendments. Or like recently, new nozomi shinkansen were introduced and added to the already tense schedule.

As uji mentioned there are many different routes.

For the route you mentioned (Asakusa-Shibuya)
the direct route from Asakusa to Shibuya, it is the Tokyo Metro Ginza Line, no transfers required its direct and costs 230yen.

Use the Tokyo Metro/Toei Subway map (found online) along with the JR East map, if you pull up a route for Tokyo that doesn't make much sense to you, Look at the Map! sometimes the most direct route or the cheapest route is listed 2nd or 3rd for Hyperdia.

When buying local short distance trains (eg subway trains etc) you don't buy a ticket from one station to another, what you buy is a fare from one station to another (if it is on the same train network), so from Asakusa to Ginza the cost is 230yen using Tokyo Metro, at the machine you purchase a ticket that has the fare of [230] on it (eg there's no button that says SHIBUYA STATION) there are maps that tell you how much it costs from one station to another.

If you needed to go somewhere that required multiple train companies, sometimes you can purchase one ticket called a TRANSFER TICKET, but sometimes you cannot, so you have to buy one ticket to your transfer station, then another ticket at the transfer station.

Asakusa to Omotesando station is also 230yen so if you wanted to get off at Omotesando instead of Shibuya that is possible as wel with your fare ticket. However lets say you buy a ticket that is only 130yen and you went to Shibuya station and that costs 230yen, there are "FARE ADJUSTMENT" machines at the final station, where you put the 130yen ticket in, then it will tell you how much you owe and you pay the difference to exit.

Also note there are 4 Asakusa Stations:
1 Tokyo Metro, 1 Toei, 1 Tobu and 1 Tsukuba Express. The Toei/Tokyo Metro and Tobu stations are near each other, however the Tsukuba Express Asakusa station is in another location. If you are using Hyperdia the Toei/Tokyo Metro and Tobu station are combined into one as ASAKUSA (TOBU), but in reality they are 3 separate stations near each other (there are signs in the area that say which is which).

See the links Uji posted on Japan-guide, if tickets are not too confusing, but if you don't want to calculate all day, getting a prepaid card such as SUICA or PASMO (or equivalent depending on which part of Japan you are in), these cards all you do is put money on them, and the money is deducted each time when you go through the fare gates, so you don't need to worry about getting transfer tickets, or purchasing single tickets each time.