If the Elf has none in Valinor (a serious problem for Elves whose kin does not live there, such as Sindar in the First Age), their king’s representative is notified: Finarfin’s for Noldor, Olwë’s for Teleri, Sindar, Nandor, and everyone else Teleri-related, Ingwë’s for Vanyar (which does not happen often) and everyone who does not identify as anything/is unsure/is mixed. Upon release, the fëar have to go through a ceremony where are given new bodies. These bodies look the same way the Elves looked around the time of their death, only undamaged. Another difference is that their hair is short: it symbolizes new life and also lets people around know that this person has been recently released from the Halls and should be treated with care.

The reembodied Elves are given a simple white shirt, a pair of wooden sandals, and a crown of dried flowers. 

(sandals)

They have to return the shirt and the shoes to Namo’s Maiar later, but they get to keep the crown. Most Elves (or, more precisely, their families) burn it in their first day of new life to mark the fact that their time as dead ones is over, but Namo’s Maiar keep reminding the public that this act is just Elven tradition and does not bring any magical benefit, and nothing bad will happen if you don’t burn it. A lot of Elves are still superstitious and do not want that crown in their house.

(crown)

Having been dressed, the Elves are led through a door and a flower arch to a meeting place where their family, friends, or the king’s representative pick them up. 

(Celebrimbor upon his release from the Halls in the Fourth Age)

In the first two cases they would go to the family/friends’ house, and in the third case, the queen provides them with a room and necessary things like clothes and items of personal hygiene.

The first few months are very tough for the newly released Elves. At the meeting place, the Maiar do lecture those picking up about the challenges that their loved one is about to go through, but it’s still hard. They have to get used to basic things like breathing, walking, feeling pain. They do not recall what the feeling of hunger feels like, so when hunger hits them, it is very scary, and they think they are dying. First hunger is also very violent, but it’s unsafe to eat any heavy foods. Elves also don’t recall that they need sleep and, feeling sleepy, again think they are dying. Some achievements of in-Halls healing might be reversed by the initial stress - and also by insensitive people. Celebrimbor’s family, for instance, is particularly “grateful” to the big-mouthed person who decided to tell him about the banner thing soon after his release. Namo’s Maiar are doing their best to educate the public, but assholes be assholes. Newly released Elves are also socially awkward and/or anxious, so there is no reembodiment party. Usually, the party happens about a year after the release, when the person is ready. They are also not supposed to work until their hair grows back. Spouses and lovers have to keep in mind that the newly released person is not going to want to have sex with them anytime soon, perhaps not until years later. Generally, taking care of a newly released person is a big challenge, but during the First Age, it becomes a universal problem, so people can support each other and share their experience. Some people make it their profession to educate families and help them take care of their reembodied members. These people are being trained by the Maiar of Irmo and Este. The kings are also supposed to provide financial support in case if caring for a reembodied person hinders a family’s financial well-being (and let me tell you, Finarfin is going to go broke). All these things are still valid in late in the Fourth Age when some Elves who died during the War of the Ring become released.