My first thought would be the HDD is going out, and yes, it can happen even on brand new systems. However, since it is so new, why are you here as opposed to contacting the manufacturer to make a warranty claim? Even if it were a refurb, you typically get 90 days, and at four weeks you'd only be at about 30 days. Make it someone else's problem to fix.
A virus isn't out of the question, but honestly it sounds like you may have purchased a system from a somewhat less than reputable source. While a virus isn't necessarily out of the question, it's also considerably less likely than a failing HDD, which an Internet speed test would not in any way be a diagnostic test for. So, I almost have to give the person credit for that rather clever bit of misdirection.
Call the shop up, tell them you would like them to replace the HDD. If the HDD in your computer is fine, they shouldn't have any problem being able to reuse it in some other build after formatting it. However, if it's bad, then they should be willing to replace it anyway if they stand behind their products. And while HDD prices have remained rather high well after manufacturer's have recovered from the floods that swept through their plants earlier this year, one HDD is unlikely to make or break a company. Especially when they can turn around, make a warranty claim with the drive's manufacturer, and get a replacement themselves.
When I first bought my PC four weeks ago and started it up, I got the following error message before restarting it again and it powered up fine. However, it has still been running as slow as my old PC, taking forever to load websites, constantly buffering each new tab and YouTube video. The Tech guy from the shop came out to look it, insisting it was fine, suggesting it might be a virus, which it couldn't be since this happened as soon as he delivered it. We ran an internet speed test, and it was fine too. This is the error message I receive on powering up:
"Reboot and select proper boot device or insert Boot Media in selected Boot device and press key"
The specs I purchased, as recommended are:
AM3+ 9700E3 (or First suggestion: Aspect FM1, ASSM-HUS motherboard)
Dual Core Processor
8Gb RAM
1Tb Hard drive
610 GT Graphics card
These are the actual specs according to my system information:
Component Details Subscore Base score
Processor AMD Athlon(tm) II X3 450 Processor 6.9 4.6
Determined by lowest subscore
Memory (RAM) 8.00 GB 7.4
Graphics NVIDIA GeForce GT 610 4.6
Gaming graphics 4096 MB Total available graphics memory 6.4
Primary hard disk 826GB Free (931GB Total) 5.9
Windows 7 Home Premium
System
--------------------------------------...
Manufacturer To Be Filled By O.E.M.
Model To Be Filled By O.E.M.
Total amount of system memory 8.00 GB RAM
System type 64-bit operating system
Number of processor cores 3
Storage
--------------------------------------...
Total size of hard disk(s) 1863 GB
Disk partition (C
826 GB Free (931 GB Total)
Media drive (D
CD/DVD
Disk partition (E
603 GB Free (932 GB Total)
Graphics
--------------------------------------...
Display adapter type NVIDIA GeForce GT 610
Total available graphics memory 4096 MB
Dedicated graphics memory 1024 MB
Dedicated system memory 0 MB
Shared system memory 3072 MB
Display adapter driver version 9.18.13.697
Primary monitor resolution 1360x768
DirectX version DirectX 10
Network
--------------------------------------...
Network Adapter Realtek PCIe GBE Family Controller
Notes
--------------------------------------...
The gaming graphics score is based on the primary graphics adapter. If this system has linked or multiple graphics adapters, some software applications may see additional performance benefits.
Caution: Based on the results of the WinSAT Disk assessment, your system drive may have slower response times under some workloads. Performance results will depend on the specific applications used and the amount of memory available. If your experience is satisfactory, no action is needed.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic