It's time for you to learn about "MOSFETS" (Metal Oxide Field Effect Transistors) or "FETs" for short. (There are a number of different types of FETs but a MOSFET is the type that you will meet and use most for power purposes.
Unlike "bipolar" transistors, which are current controlled devices (more on that later maybe), a MOSFET is a Voltage controlled device. As an approximation, the current through a MOSFET is controlled by the voltage on its "gate" terminal. While you can in fact get bipolar transistors that allow such a high ratio of controlled to controlling current the MOSFET does away with this by effectively not needing any current at all for control, because it is voltage (and not current) controlled. Practical aspects mean that there is some current flow involved, and significant current may be involved for small periods, but this is mainly to charge and discharge the substantial capacitances involved with the MOSFET's gate. All this is important but is not essential to a first understanding.
I have used a search engine to look for "MOSFET tutorial". The references below in  are only a few of the many thousands of sites that will tell you more about MOSFETS. The ones I have shown look like a good start.
MUCH more is available but these will give you a good start.
Armed with this knowledge you can venture out into MOSFET land.
A very important aspect is that the MOSFET that you choose is able to be turned on and off OK properly by the available Voltage. You may have about 3V or about 5V available from your controller - maybe more. Traditionally MOSFETS needed typically about 12V to control them well. You can get "logic gate" MOSFETS that need only 5V or even less to control them. This voltage is known as the "gate threshhold voltage". It is applied to the gate terminal relative to the source terminal. (For an N type MOSFET the gate needs to be more positive than the gate to turn the FET on. You will understand N & P type FETs and more after looking at the turotials so I will not talk about such details).
The threshhold voltage may be named Vth or Vgs_th or Vgs_threshold or similar. Selecting a FET that will be easily turned on by the available gate voltage is a "good start" (ie essential). Vth is the voltage where the FET is just turned on and a very small amount of current can flow. Choosing a Vth which is 1 to 2 volts less than the available control voltage is necessary if you want the FET vto be able to be fully controllable. When the available control voltage is only about 3 Volts there are relatively few MOSFETS available which have an adequately low Vth. The example that I give below was selected by using Digikey's selector guide using the parameters listed below and then asking for the lowest cost MOSFET in single quantity which met the spec and which was in stock. To my great surprise the very cheapest MOSFET listed was utterly superb for both experimenting and for general use. This device has far far far better specs than most you will encounter. It's worst case turn on voltage is very low, it's on-resistance is also very low and its Voltage and current ratings are impressive. if you are in the US or have access to Digikey then this part is worth trying. You will have trouble finding a better one for most applications of this sort.
I could write a few more pages here easily BUT the many many available tutorial and data sheets will do a better job.
To start, here is how to find some devices that are available in the US and which wil do what you want. Tell us where you are and what sort of $ you are happy to spend and we may give other additional advice. A power MOSFET able to do what you want can cost well under $US1. Or a lot more in some special cases.
If you go to www.digikey.com and search for
MOSFET N Channel TO220 logic
and then choose "FETS single"
you will find many experimenter friendly devices.
Cheapest here at $US0.94 in 1's and in stock is IPP096N03LGIN
Don't be scare by the long part number
This is an amazingly capable part for the money.
1.2V MAX gate threshold voltage - very good and very important.
30V rated at 35 A !!!
10 milliohm on resistance when properly driven.
A superb start.
Many more there ...
 MOSFET tutorials and similar.
(a) Wikipedia introduction looks good
(b) This is page 1 of a series. MOSFETs start on page 6, but reading 1 - 5 will help you understand bipolar traansistors and the differences involves.
From start http://www.electronics-tutorials.ws/transistor/tran_1.html
(c) IR (International Rectifier) are a very large and reputable MOSFET maker. This introduction is at a higher technical level than some other material but is extremely good
(d) These people are less known, but this medium technical level introduction also looks good.