The Requests posted on the bulletin board can result in rewards that vary from tools and other items indepensable to your character to small amounts of money and sometimes nothing more than a simple 'thanks' from the originator of the Quest. You do need to fulfill all Requests, I expect, in order to be able to move on to new ones. Characters tend to give the same kind of reward as well. A Character who gives seeds as a reward will continue to give seeds. A Character who gives tools that help with Monster Care will continue to give items that relate to Monsters (hint, hint!). I rather suspect that there are some time requirements for quest appearances as even after completing every quest on the board, it was a day or two before a substantial number of new requests appeared. In fact, I had a barn filled with monsters, the Milker and the Shears for a few days before the Request that resulted in the Brush appeared on the board.
Again, patience is required sometimes in every Harvest Moon game. The heart levels of my monsters did not increase much until I had the Brush but it was not a real hardship to be obliged to wait a couple of days.
One suggestion I would make to any player who begins to collect monsters is to grow as much grass as possible in your field in order to be able to feed them without going bankrupt. The cost of fodder is prohibitive to a beginning farmer. A plot of grass grown in your field can produce 36 portions of fodder, not to mention 1 Rune Point. Grass will continue to grow after it is cut as long as you water it.
One bag of Grass Seed costs 300G. One portion of Fodder or Feed costs 400G. In the initial season of the game, if you have as few as four monsters and no ripe grass for fodder, you will be forced to spend 1200G per day on Feed. If you have your barn filled with monsters, 16 monsters will require 16 fodder per day, making the cost 6400G per day if purchased. By investing in Grass Seeds, you save a significant amount of money as well as creating another source for Rune Points. If you stagger the planting or reaping a little, you can make certain that one or two plots of grass always are ripe each day, giving you constant access to 1 or 2 Rune Points from grass alone. (Reaping ALL your grass on the same day in the early stages of the game will give you more Rune Points than you will be able to use effectively. It is better to stagger the harvests.)
Another aspect of farming needs to be mentioned. If you keep fully-ripened crops in any field (whether it is your own field or a dungeon), you must continue to water them regularly. Otherwise, they will begin to die. By losing the crop in even a single square in a 3 x 3 plot, you lose any chance of having that plot produce Rune Points. In other words, if you neglect any plot and even one plant dies, you might as well harvest the remaining crops in that plot and plant a new Crop.
A new aspect of the game is Music. The Music component in RF2 reminds me a little of the flute and the ability to use it to train your Dog to perform various moves in 'Save the Homeland'. Music in Rune Factory 2 is used, inter alia, to interact with tamed monster companions in battle. Your first musical 'ability' is a 'Special Concerto', described as 'A sprightly tune. Allied monsters will flash yellow and perform their best move if successful.' As each type of monster has unique abilities, the 'Special Concerto' will have different effects depending on the identity of the Monster who accompanies your character.