A new HDD is no guarantee that your computer will run "fast". That said, if your system was build in the last 15 years, it should surf online just fine. Sounds like you may be riddled with viruses. In that case, go ahead and get a new drive...install your OS there, and *carefully* migrate any files you must back over. Makes sure to install an anti-virus/anti-malware program immediately.
Between regular updates, using anti-virus, and "smart browsing"...you shouldn't have an issue.