Nasturtiums, like these Vesuvius heirlooms from Burpee , are a good choice for beginners. If you do have mice, as can be a problem in more rural areas or with older houses, try to find the top of a book case or something tall that stands away from the wall a bit that they can't climb. Or, harvest in the morning on a very sunny day and let seeds dry outside until late afternoon. If you are new at seed saving, certain varieties are easier to save than others. Large seeded flowers such as nasturtiums and 4 o'clocks are easy to begin with.
Marigolds are also simple to dry and offer a high success rate for beginners.
Large seeded vegetables like pumpkins and edible seeds like sunflowers also work well. Do spread the seeds out separately, as sunflower seeds left in the seed head are more at risk of mildew also. Saving seeds from a squishy fruit or vegetable - scoop seeds out onto a layer of paper towels and press off extra moisture before setting to dry. Tomato and cantalope seeds benefit from this, for example. This saves startup costs the following year.