In a civil action: If you are serving written discovery on a party, under the federal rules of civil procedure, or in any state whose adopted the model rules, (rules 33-37 typically), you need only send your requests for interrogatories, the production of documents, or for admissions to the party (through their counsel unless they are pro se), accompanied by the discovery notice, detailing the time frame in which they are due. It is typical that this time frame will get extended, sometimes by double or triple the original 21 day period plus 3-5 days for mailing.
When the party is represented (this is 99% of the time) the attorney will just call you or email you for an extension and it is pro forma to assent to this. The court will not be happy if they need to seek permission of the court for this because you refused, as it is that common, and you will just get a reputation as a non-cooperative pain (not to mention, the time will come when you need an extension, and what comes around goes around). If they seek an unreasonable amount of time (I'm talking so close to the discovery cut off that you'd not have time for depositions after receipt), then you can limit it, but a couple of months is common. This need not go through the court. If they are not represented, they may just be late rather than calling, in which case you have a duty to contact them before involving the court. This is just because they don't know it is the norm, and it is nearly impossible to pull together everything needed from a full set of discovery in 21 days.
When there is a true discovery dispute, you must show the court that you made all due diligent efforts to resolve it on your own. I don't have my rules in front of me (I will edit and add this when I do) ... but this is in the rules, it is not just practically speaking. Also, many jurisdictions require that you contact the court's clerk and get permission before filing any discovery motions. Sending a subpoena is not ever how this is dealt with. If the opposing party fails to respond, or if they are continually late and despite numerous contacts they still do not produce, then you would schedule a discovery conference with the court at which time you will seek a motion to compel. The court will typically give them even more time at this point.
You are correct in that when a third-party is served with discovery request, since they have no obligation to take part in the case but for the presence of a subpoena; hence, in this situation you would use a subpoena to request whatever it is you are requesting. They may get their own lawyer who will try to limit your right to get anything from them as a non-essential party, but if it is really relevant you can do it. You would also use the subpoena for depositions, to ensure that witnesses present themselves. Otherwise, subpoenas are reserved for acquiring witnesses to appear in court or for custodians of records to bring records to court for hearings or trial.