When the news announced the US governmental shutdown today, it was also mentioned that tourists might be affected as well. The example that was given is that the national parks will be closed. Is that it, or will I notice other effects, such as no public transport, longer delays at the airport, etc?
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You shouldn't notice much impact on the actual getting around part of travel.
Public transport, as well as things like highway maintenance and management are all done at the state level, and are unaffected by short term federal government shutdowns. Similarly, the TSA, FAA, and Border Patrol are all continuing to operate, with their employees being paid on the resumption of government activity. AMTRAK is a privately owned enterprise; while they receive federal subsidies, they should be able to continue operating at full capacity for the foreseeable future.
Where you will notice a lack of activity is in trying to use other government services. As a traveller, the primary impact you'll see here is, of course the National Parks Service. This means parks and various monuments and historic sites are closed. This includes the Smithsonian museums in Washington, DC. Do keep in mind however, that much of the nations parkland, as well as many historic sites and monuments and museums are not federally maintained or operated, and at those facilities, it should be business as usual.
While Visa processing will be delayed or affected, it won't be completely shut down. For the most part, as a visitor, you shouldn't notice much impact from the shutdown. If it drags out for more than a few weeks, the lack of available funds could cause larger issues, but for now, as a foreigner, you can safely not worry about it.
(N.B. Because of its peculiar governmental status, visitors to Washington D.C. specifically might experience a larger impact than visitors to other cities. In particular, most sites of tourist interest in DC are federally operated and will be closed. YMMV.)
For further details, CNN has a comprehensive and searchable list of which government agencies are experiencing furloughs or shutdowns, and to what degree. You can consult it if there is any particular federal agency with which you need to conduct business.