Nexus posted some really good information. I'm skimming so I apologize if I missed something, but I would suggest--whether you go laptop or desktop--to try and get a system with a solid state drive (SSD). If you're buying new, a SSD is a must. The main benefits are less heat (great for laptops), faster boot times, and faster program access, while the main drawbacks are cost (compared to traditional hard drives).
For example, I have a quad-core Acer laptop that is supposed to have 1 GB of dedicated video (Acer Aspire AS5560G-Sb468), but it's running on a 500 GB hard disk drive (HDD) rather than a SSD and it just can't compete with the cooling of my desktop computers. I also found I did not like the brightness of the Acer screen, and I turned it down all the way I could. But like Nexus said, make sure to have at least 8 GB of RAM. If you go further than that, you need to make sure your computer and the operating system will allow you to go farther. Use
Crucial for help when finding compatible parts.
I made the computer I'm currently running about a year and a half ago, so about January 2014. Everything included cost about $1,000 (monitor not included), for what is a 4.0 GHZ octa-core computer w/ 16 GB RAM, Windows 7 OS, and 2 GB video. My main drive is a 120 GB SSD, and I have two HDDs with 1 TB each. At this point, I'm looking into cloning the 120 GB SSD onto a 512 GB or greater SSD; SSDs continue to fall in price. That's perhaps the best thing about a desktop computer, in that you can add more drives and bays easier. You can't do a whole lot of customizing with a laptop. But as Nexus said, consider that when you add a new graphics card or anything else to a stock desktop computer purchased at a store on online, you may very well need to replace the power supply unit.
If you do go laptop, I'd suggest ASUS. They seem like one of the better brands, and it seems like they have models with the newer hybrid drives (HDD and SSD in one). If you go desktop, depending on where you want to start, either go with a pre-made system that has a good interface--IE, USB 3.0 ports on the front of case that you like--or think about case size, like mid-tower or full-tower. If you don't feel comfortable building the whole thing, consider looking into only replacing the power supply unit (PSU), the RAM, and adding a dedicated video. Use crucial for the RAM in that case, find a compatible graphics card (PCI-E x16 most likely), figure out if you'll need the graphics card to be low-profile (if you have a lot of need for cards in the back, like for wireless, etc.), then find out what kind of PSU wattage the graphics card will need. Then, examine the current PSU, and count up the connections you'll need to replace so that your new one will be able to connect to all of the devices you need to connect to.