In regards to re-signing players, there's different groups of players on your roster. And for the most part, how you structure their contract is dependent on their service time.
Most of the contracts you'll need to manage each off-season are your minor leaguers. You can unilaterally renew their contracts. You can renew their contract for no less than 80% of their previous year's salary and they have to accept. For these players, you do not have to give them what they're asking for. You'll need to tender them a contract by a certain date in order to ensure they don't become a free agent. This is usually December 2nd if I recall correctly.
Once a player has reached 3 years of service time, he becomes arbitration eligible. Whether you offer this player arbitration or not will be dependent on various things. You'll need to weigh how much he's asking for and how valuable he is to your team. For larger budget teams, the difference in what you want to offer and what the player wants probably doesn't break the bank. But for smaller market teams, it could be a factor.
For arbitration eligible players, you'll have a deadline to offer it to them. At this time, you'll see what the player is asking for. And you'll be required to submit your offer. Based on their performance, you can probably get a good idea on if they would win their case or not.
Some of these players may be worth signing long-term. A player becomes eligible for free agency when they reach 6 years of service time. For example, last year for me Michael Pineda had reached 4 years of service time. I had one more off-season to sign him long-term but I ended up giving him a 5 year deal. Ivan Nova had reached 5 years of service time. It was my last chance to sign him long-term without him being exposed to the open market. However, he had been inconsistent for me and I wasn't ready to commit to him long-term. I wanted to see how he does this season. And I figure I will be content either way next off-season to either re-sign him or tender him a qualifying offer. If he leaves, I'll probably be fine with the draft pick compensation.
So getting into tendering qualifying offers. Any player eligible for free agency that spent the entire year on your 25 man roster can be tendered a qualifying offer. A qualifying offer is equal to the average of the 125 richest contracts in baseball. In my seasons, the offer has been 9M and 11M I believe. So, you don't just give this offer to anybody that is eligible for free agency. You extend this offer to a player you don't want to lose. If you tender a player a qualifying offer and he signs with another team, they forfeit their 1st eligible draft pick to you. The top 10 draft picks are protected. In this case, they would forfeit their 2nd round selection.
So with that in mind, you should be aware of this when signing free agents yourself. Some free agents will be must signs and worth giving up a draft pick. Others may not be worth losing that draft pick. And you forfeit a draft pick for every qualified free agent you sign.