Hi  Sebastiano,

My understanding of what you want is:

- When the first contact is created in an account, you want the account to have the "First_Contact__c" field point to that contact.
- When the first contact is deleted in an account, you want the account to have the "First_Contact__c" field point to the now oldest contact in the account.

If ths is right, the trigger you want to write is not an Account trigger, but a Contact trigger.  In that trigger, have the "after insert" look at the account field First_Contact__c to see if it is null, and if so, update it with the new id. Have the "after delete" look at the account field First_Contact__c to see if it is the current record, if so, get all contacts of all accounts where this is true, sorted by account id then create date, and loop through this list to find which contact, if any, should be the new "first contact" for each account.

If you already have accounts and contacts and need to seed the initial data, I'd suggest you write a throw-away batch class to do it.

Also, it is a "best practice" to have the trigger code delegate the actual work to a class, like in the following:

trigger ContactTriggger on Contact (after insert, before delete) {
    ContactTriggerHandler handler = new ContactTriggerHandler();
    if (Trigger.isInsert && Trigger.isAfter) {
        handler.OnAfterInsert(Trigger.new);
    }
    if (Trigger.isDelete && Trigger.isAfter) {
        handler.OnAfterDelete(Trigger.old);
    }
}

Here is the trigger handler code I'd suggest:
 

public class ContactTriggerHandler {

    // if the contact is the first Contact added to an account, set the "First_Contact__c" field in that account
    // Note, if you're inserting multiple contacts for an account that currently has no "First_Contact__c" set,
    // an arbitrary one will be used.
    public void OnAfterInsert(List<Contact> items) {
        Set<Id> accountIds = new Set<Id>();
        for (Contact c : items) {
            if (c.AccountId != null) {
                accountIds.add(c.AccountId);
            }
        }
        // get entries that still need a "first contact" to be set
        Map<Id, Account> acctMap = new Map<Id, Account>(
            [SELECT id, First_Contact__c FROM Account WHERE Id in :accountIds AND First_Contact__c = null]);
        List<Account> accountsToUpdate = new List<Account>();
        for (Contact c : items) {
            if (c.AccountId != null) {
                Account a = acctMap.get(c.AccountId);
                if (a != null && a.First_Contact__c == null) {
                    a.First_Contact__c = c.Id;
                    accountsToUpdate.add(a);
                }
            }
        }
        // set the "First_Contact__c" fields in accounts that needed one
        update accountsToUpdate;      
    }
   
    // if deleting the formerly "First contact" in an account, find the next oldest contact to make the "first contact"
    public void OnAfterDelete(List<Contact> items) {
        Set<Id> accountIds = new Set<id>();
        for (Contact c : items) {
            if (c.AccountId != null) {
                accountIds.add(c.AccountId);
            }
        }
        Map<Id, Account> acctMap = new Map<Id, Account>(
            [SELECT id, First_Contact__c FROM Account WHERE Id in :accountIds]);
        Set<Id> accountIdsWhereWeDeletedFirstContact = new Set<Id>();
        for (Contact c : items) {
            if (c.AccountId != null) {
                Account a = acctMap.get(c.AccountId);
                if (a != null && a.First_Contact__c == null) {
                    accountIdsWhereWeDeletedFirstContact.add(a.Id);
                }
            }
        }
        List<Contact> remainingContacts = [SELECT id, AccountId FROM Contact WHERE AccountId in :accountIdsWhereWeDeletedFirstContact ORDER BY AccountId, CreatedDate];
        acctMap = new Map<Id, Account>(
            [SELECT id, First_Contact__c FROM Account WHERE Id in :accountIdsWhereWeDeletedFirstContact]);
        // in case we have no more contacts for the account, set First_Contact__c to null
        for (Account a : acctMap.values()) {
            a.First_Contact__c = null;
        }
        for (Contact c : remainingContacts) {
            Account a = acctMap.get(c.AccountId);
            if (a.First_Contact__c == null) {
                a.First_Contact__c = c.Id; 
            }
        }
        List<Account> accountsToUpdate = acctMap.values();
        update accountsToUpdate;      
    }
}