Federal courts vs. state courts..?

Is it true that State courts will take any subject matter, including federal issues of US constitution, statutes, and treaties, but WON'T take bankruptcy, tax and patent/trademark issues? and Federal courts will only take federal issues? Thanks!
Answers:
That's not exactly right.
State courts are courts of general subject matter juristiction. That means that unless there's some specific area where juristiction has been taken away, a state court can hear the case. The three areas you mention, bankruptcy, patent/copyright, and federal tax are areas where the states gave sole juristiction to the federal courts.
Federal courts are courts of limited subject matter juristiction. Juristicion must be granted to them from the states, or come from the constitution. Therefore, federal courts have juristiction over federal questions: issues that arise out of the constitution or federal law.
They have also been granted juristiction for diversity cases, where the amount in contoversy is over $75,000. Diversity cases are where the parties in different states. This was to ensure fairness in such actions. The specific rules get complicated.
So it's not true that federal courts only take federal issues, they can take other cases too, through diversity.

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No, that's not accurate.
There are many topics that can be filed in EITHER state or federal court. There are only a few that MUST be filed in federal court.
The three you listed -- bankruptcy, tax and patent/trademark -- as well as copyright, or any claim that rests solely on relief provided by a federal statute.
Most suits are based on state law claims, because most laws that would apply on a regular basis are state laws. And if there is a valid state law claim, any secondary federal claims can be filed at the same time.

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The famous case where the White House made representation in support of Anna Nicole Smith in Marshall v Marshall where the decision was in the favour of Ms Smith (aka Ms Marshall). The motivation of the White House was not actually to support her inheritance case against her step son, but to ensure that judges support the right of jurisdiction of Federal courts in any cases covered by Federal law regardless if Federal courts choose to relegate any actions or decisions to state courts. That is, Federal courts have the jurisdiction which they normally do not wish to exercise but relegate to state courts.
Obviously, if you violated state taxation laws, you would be prosecuted thro the court of the state whose laws you violated.
If you violated taxation laws of Colorado by making statements defrauding the Colorado government from the state of Nevada through mail, telephone and email, you would fall into the Federal laws of wire, mail and electronic fraud. The Federal government and subsequently the Federal courts could choose to act or allow and assist the state actions to take their course.
In civil cases, if you wish to take actions against parties where the harm was allegedly committed against you across, say, three states - would you choose to take action at all of three states, or rely on Federal laws and statutes and take action only at a Federal court or even at both state and Federal levels? That depends on your being able to afford a wide spread action, the monetary amount of loss and harm and what you wish to focus on in order to have the most factors aligned to your favour.

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State courts are courts of general jurisdiction, meaning they can generally hear any claims with a few exceptions such as the ones you mentioned.
Federal courts can only hear cases in which they have subject matter jurisdiction over. A court generally has subject matter jurisdiction over cases where (1) there is complete diversity of the parties citizenship and the amount of controversy exceeds $75,000 or (2) the claims arise out of a federal statute or the US Constitution.

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Your understanding is generally correct. Federal Courts will sometimes hear matters of state law if they are involved in cases where federal law is also at stake. Federal Courts also hear cases where residents of one state are suing residents of another state and the amount in controversy is over $75,000. If it isn't a question governed by federal law and its not a suit across state lines for more than $75,000, then it probably belongs in state court.
And yes, matters like patent and bankruptcy are governed by federal law.