Hmmm, I doubt it's your Internet connection because usually your network adapter is the last thing to start on a bootup. Unless of course you have Boot from Lan enabled in your BIOS, which by default is usually OFF, so unless you specifically turned it on, chances are it's off. In addition, there is no way to have the network adapter only come on when you open your browser.
One thing you can try is go to START > RUN and type in: "msconfig" without the quotes and hit Enter.
This will bring you to your System Configuration Utility. Be VERY CAREFUL about changing things here because it could screw up your system.
Go to the STARTUP tab. All the items with green check marks are things that are starting up when you boot up your computer. Chances are, there are a few things in there that you don't need.
If you could list those items here, mainly just the 'Startup Item' name and the location (for example: C:\Windows\System32\) I could let you know which you can uncheck.
If you're not keen on typing that list, you can take a screenshot, post it here, scroll down some more and take another screenshot, etc.
Another thing that could be causing a slow boot are Services. You may have several services starting that don't need to be starting. By default, Microsoft enables several completely unnecessary services that are resource hogs. You can check this list by going to START > SETTINGS > CONTROL PANEL > ADMINISTRATIVE TOOLS > SERVICES
If you can type our or screencap the list of services, I can let you know which we can disable.
Finally, the third thing that could be affecting your startup time is hardware. Do you know how fast your processor is? How much RAM you have? How much space you have left on your C: drive?
We can fix this Bossy and get your system starting up much faster.
