At a HGI:
At Hilton Garden Inn hotels, Gold VIP members will receive their choice of either 750 HHonors bonus points OR 2 bottles of water and complimentary Full American Breakfast for member and one guest throughout the stay and upgraded room accommodations (subject to availability upon check-in).
As many of the Garden Inn's don't have different types of rooms to upgrade you to, it's probably going to be the points or the water (which you may have to ask for) and breakfast. Occasionally you may luck out and get a suite if it's a HGI and suites, but don't count on that.
Upgrades can happen at Web Checkin, but remember upgrades are not guaranteed by status at Hilton. I believe once you start the process it will offer you ranges of different rooms if you have a selection. You can not pick a specific room though, so it may be hit or miss.
From their description of web checkin:
Available within the fourth quarter 2004 at selected hotels across all brands, Web-based check-in will enable Gold and Diamond Hilton HHonors(R) members with password-protected online accounts to check into their hotel rooms in advance of arrival, regardless of how they made their reservation. Guests simply access their reservation via one of the brand websites, select a room with features that best meet their needs, and print the confirmation. Since guest information is stored within the online personal account, the hotel does not need to collect the same information at check-in.
Once the guest checks in remotely via the Web, the hotel is notified electronically of the pending arrival. The hotel front desk then completes all pre-arrival and check-in processes so that the guest's key card and registration packet are ready and waiting for the guest's arrival. Upon arrival at the hotel, guests exchange the check-in receipt for their prepared room key and welcome packet, which will be waiting for them at the hotel front desk.
I don't believe you can cancel a web check in, may be wrong on that one.
Even with web checkin, you can ask at the counter if there's a better room available.