I would think that the hotel probably has a courtesy service that would take you as far as Central Square, at least. If not, a cab from the Hyatt wouldn't be that expensive, especially if there are more than 1 of you, relative to the cost of multiple subway tokens. The subway is $1.25 when boarding at an underground station... the fares go up above ground for inbound service, and are free outbound, when boarding above ground
You could walk to Central Square and take the Red Line of the "T" (subway) to Park St., where the Freedom Trail begins, or you could walk along Memorial Drive to Mass Ave and either take the "#1 Bus" from MIT across the bridge to Boston, or just walk across the bridge on foot (if it's a nice day). If you walked in the other direction along Memorial Drive, you could walk across the "B.U. Bridge", then catch the Green Line inbound to Copley Square or Park St. Any of these walks would be a mile or so... the walk across the Mass Ave bridge probably being the longest option. The walk to Central Square being the least "scenic".
Near Copley Square, you could get a Boston Duck Tour or stroll around the many shops in the area. It's a pleasant walk from the Copley area through the Public Garden to Charles St., Beacon Hill, then down the other side, through Government Center to Quincy Market... where you could pick up the Freedom Trail. Bear in mind that August can be very hot here...
I've never used a Go Card, but I've heard nothing but good things about them.
The hotel isn't in a good location for public transportation. It's also not a great location for walking around as it's isolated from most of Cambridge, on a very busy road, and isn't an easy walk to the two nearby bridges into Boston. Your best bet would probably be to take a cab. If there is a shuttle to the Kendall or Central Square T-stop then you can take that easilly to anywhere you'd want to go.
Check out mbta.com for more information on public transportation.
I work by the Cambridgeside Galleria and have seen a shuttle vehicle from the Hyatt dropping off or picking up people there. Since the Kendall stop on the Red Line is en route, I'm sure getting a lift there would be a non-issue.
There's also an express bus route called the CT2, which charges the regular 90-cent fare and stops on Amesbury St by Memorial Dr right outside the hotel. Boarding on the hotel side would get you delivered to within an easy walk of Fenway Park (get off at either Beacon St or Brookline Ave/Park Dr) or the MFA and Gardner Museum (Longwood/Huntington stop.) You can also connect right to either the C trolley on the Green Line by walking west on Beacon, or the E trolley at Huntington. Boarding on the opposite side of Amesbury St, you can ride the CT2 to Mass. Ave, on to Kendall Station, and then to Inman and Union Squares before looping back around to Sullivan Station on the Orange Line. This bus starts running in the early morning but has its last run from either terminus at 7 in the evening. You can't beat it for when you want to save up your walking energy.
I've often seen their shuttle in Boston, but don't know what the arrangement is. Scheduled? On-demand? Business groups or conventions? They conspicuously don't mention it in their website. Try calling them directly, and asking the concierge. Seems they would have the service, in order to remain competitive with the downtown hotels. Even if it takes persuading someone at the hotel to take you in, this would be better than the other options, if it's possible.
The Boston Go Card provides excellent savings. All sorts of attractions are offered many of which you will attend or use anyway. Check their Website. They also will provide maps. Also, they offer a discount to the normal price which I believe was offfered this month.