The most common form element are input, checkbox, radio buttons, textarea and dropdown boxes and I will try to cover them all, even though the code might be the same or very much alike.

Input boxes

Fetching content is done with:

// fetching the value
$("#inputbox").val();

Try it:
Make a popup with the contents

Setting new content is done with:

// setting a new value
$("#inputbox").val("We have now changed the text");

Try it:
Change the content to “We have now changed the text”

Check boxes / Radio boxes

Both check boxes and radio boxes works the same way.

Checking all fields:

// loop through each checkfield
$("#elementID input:checkbox").each( function() {
  // set checked=checked
  $(this).attr("checked", "checked");
});

Try it:
Check all boxes

// loop through each checkfield
$("#elementID input:checkbox").each( function() {
  // if field is checked, then uncheck or the other way around
  this.checked ? $(this).attr("checked", "") : $(this).attr("checked", "checked");
});

Toggle boxes

Outputting the checked status for each field:

// temporary variable for output purposes
var status = "";
// loop through each fieldfield
$("#elementID input:checkbox").each( function() {
  // save field value and checkstate in variable
  status += $(this).val() + " is " + this.checked + ",  ";
});
// make a popup box with the result
alert(status);

Try it:
Output checkbox status

Textarea

Textarea sample text
Fetching content is done with:

// fetching the value
$("#textarea").html();

Try it:
Make a popup with the contents

Setting new content is done with:

// setting a new value
$("#textarea").html("We have now changed the text");

Try it:
Change the content to “We have now changed the text”

Dropdown boxes

Fetch the value if the dropdown box:

$("#selectbox").val()

Try it:
Output which option is selected

Selecting option 2 in the downdown box:

// loop through options
$("#selectbox option").each( function() {
  // if the value is equal to 'select2', then select it
  if ($(this).val() == "select2") {
    $(this).attr("selected", "selected");
  }
});

Try it:
Select option 2

Selecting option 3 in the downdown box:

// loop through options
$("#selectbox option").each( function() {
  // if the value is equal to 'select3', then select it
  if ($(this).val() == "select3") {
    $(this).attr("selected", "selected");
  }
});

Try it:
Select option 3

There you go. A small demonstration of how to access form elements. There above examples is just one way of doing it. There are other ways as well.