There is a number of organisations which are listed by the US government as terrorist organisations - Wikipedia
What are the legal implications of this in relation to the 1st Amendment? If, for example, there is somebody who is non-violently expressing sympathy for the goals of an organisation declared as terrorist by the US government, is that illegal in the US?
The U.S. Department of State maintains the list of Foreign Terrorist Organizations (FTO) and along with other government departments are charged with the process of making said designations. Under U.S. Law, it is illegal to provide material support to any FTO with Material support being defined by 18 U.S.C. ยง 2339A(b) as:
currency or monetary instruments or financial securities, financial services, lodging, training, expert advice or assistance, safehouses, false documentation or identification, communications equipment, facilities, weapons, lethal substances, explosives, personnel, transportation, and other physical assets except medicine or religious materials.
Note that the list exempts religious materials, such as holy books or documents speaking of ideology that may motivate the group. This would serve to make clear that the law is not against expression of sympathy for the ideology, but is targeting the means by which the FTO can use violence in support of their belief.
It should also be noted that the law bars entry to the United States by aliens who are known members or representatives of an FTO, but this is a higher level of Material Support as it implies that such people are actively engaged in an FTO's activities.
Finally, the law states that financial institutions that operate in the U.S. that become aware of accounts belonging to FTOs should take control of the accounts and value within it and immediately report to the treasury.
An FTO is by definition any organization that participates in activities that meet the legal definition of terrorism, which is neutral towards the beliefs motivating such activities and instead focuses on illegal actions.
I guess you will be protected by 1st amendment (freedom of speech) as long as you only express support for such a group. If, however, you act in support of such a group, e.g. by collecting money for them or involve in arms trading, you'll be on the hook.
On the other hand, it also depends on what exactly you say. If you sympathize with such a group and call for violence in it's name, this may or may not be considered a crime on it's own (in particular if somebody really acts violently after your speech). See e.g. this question. At the very least, I would expect that this would raise a suspicion against you and cause an observation by the police.
