It differs by department, especially between rural and urban areas. Most rural departments will use the county jail instead of a city jail. Most of the large cities have a city facility for the initial hearing.
When an officer decides to arrest you, he will let you know (legally must identify himself, his agency, his authority, and the reason UNLESS it is obvious so figure he says you are under arrest). Most officers now cuff as soon as they decide you are under arrest and then search you, a change from older tactics where the search was before cuffing. At the scene, the officer will normally only take evidence or obvious weapons from you.
Officers, as TXInvestigator pointed out, are only required to read you your rights if they intend to ask you questions about the crime. Many officers will read them to you anyway, but many will not. If you make a statement on your own (not in answer to a question) without being read your rights, it is admissible. Most crooks think they must be read their rights, so they say some dumb things thinking I cannot use it. I have even had some tell me that they are not really under arrest because I did not read them their rights.
You are transported to a booking facility. You get turned over to the facility for processing (prints, paperwork, another search, etc.) and the facility will have its own rules for what you get to keep. Most places take all of the loose personal property but only take belts and shoelaces and the like if you are under a suicide watch. The property they take can be kept in the facility property room or the officer can take it to the police station property. If it is valuable, plan on it going to the police as the jail will not want the responsibility.
Legally, there is no such thing as a standard bail when you are booked. In reality, most cases have a set bail amount but it is not applied until you see a judge and he sets it. The judge at the initial hearing can set almost any bail he wants. Large cities usually have gone to a 24 hour judge for this hearing. Smaller towns and rural areas will make you wait until the next day (sometimes the next business day, they can legally go up to 72 hours if it is a weekend). BTW, the latest version of the law lets them use video conferencing for the hearing or take you to another county than the one you were arrested in.
Not sure what you mean by an I-bond, but assume you are referring to personal recognizance bonds. A judge can order you to pay a cash bond, a surety bond, or be released on your signature (personal recognizance). Most counties have a procedure where the bond office in the jail can determine if they want to let you out on a PR bond if the judge did not specifically order otherwise.
Assuming you are having to actually post a bond, you cannot use your own money or credit cards because you no longer have them in your possession. You can sometimes work with the jail to get them, but plan on having someone from the outside post your bond. If the judge allows a surety bond, say for $10,000 as one example, you can post this in several ways. You can actually pay the court the money, which you will get back when you show up for the trial - even if convicted. You can sign a lien on some property worth that much, like your house. Or you can use a bail bondsman to post it. In the last case, you generally pay between 10 and 25% and he posts the cash (well, he has already posted liens on property and has an amount he is allowed to post). You are out the money you pay him, basically it is interest on the loan of his money for bond.
Contrary to the movies, you are not restricted to just one phone call. In most cases, you can make as many as you want but they will all be collect calls. And the fees are outrageous as cities and counties have learned they can make a profit on these calls.
When you are released, the jail will give you the property that was left with them. For the property that was taken by the police, you need to go to their property room, which is usually a Monday through Friday 9 to 5 thing.
If you were driving a car at the time, it can depend on many circumstances. The officer is legally responsible for the car and its contents, so he can search (inventory) it and impound it at your expense. If you are willing to waive the legal responsibility, he may let someone else drive it or even leave it legally parked at the location. This will depend on local rules, the exact situation, if he thinks there is a reason to search the car, and your cooperation level. It is up to the officer in most cases, departments will allow other things but I have never heard of an officer being in trouble for going the safe route and impounding a person's car.
If you have minor children with you, the police will take custody of them. In most cases, they will call for another relative to get them, but only if it is a reasonable amount of time. They can take them to CPS or to a local children's center if no other adult is available. A lot of this will depend on local rules also. For example, San Antonio has a juvenile detective on duty all the time, so there will be someone available to watch while they wait for an adult. Luling has two officers, so they cannot usually wait nearly as long with the child (one is taking you, the other has to handle calls).
If you get arrested in a large city, figure on 4 to 6 hours until you can get out. If it is a rural area, figure on at least noon the next business day.
BTW, one of the meanest things a cop can do to you is take your wallet and put it in the property room. When he does, you need ID to get it, and all of the ID is in the wallet. How do you prove who you are for the jail when you wallet is in a building across the town or county and you cannot get it?