First of all, we need to learn that to form the Perfect tense in French we will need to learn the present tense of the two following verbs: avoir (which means to have) and être (the verb to be). The two verbs act as the auxilary verbs in the Perfect tense in French.
Secondly, we need to decide whether the verb we're dealing with is an 'avoir verb' or an 'être verb'. The main rules to remember while learning which verbs go with avoir and which verbs go with être are: 1, être is the auxilary verb of all movement verbs such as aller (to go), descendre (to go down) or tomber (to fall); 2, être is also the auxilary of all the reflexive verbs such as s'assoeir (to sit down), se coucher (to go to bed) or s'inquiéter (to worry); 3, all other verbs use avoir as an auxilary.
Thirdly, we need to learn all the past participles!
Many of them are irregular so we just have to sit down and learn them. But it's all not so bad because there are patterns thanks to which we can group certain verbs together which faciliates the learning. For example, many past participles end in 'u': voluloir (to want) - voulou, boire (to drink) - bu, voir (to see) - vu, courir (to run) - couru... Others end in 't': faire (to make or do) - fait, écrire (to write) - écrit, produire (to produce) - produit. Some end in 's': prendre (to take) - pris, reprendre (to take up/ start again) - repris, s'asseoir (to sit down) - s'assit.
The regular verbs are much easier to learn - take off the infinitive ending (ER/RE/IR) and add é, eg. toucher (to touch) - touché, bavarder (to chat) - bavardé etc.
One more important thing about past participles - there are two occasions on which we have to agree them with the person who carried out the action. To do that we add an 'e' (sometimes and extra 'e' ending up with two 'e's at the end of the word) if the person who carried out the action was female; or an 's' if there was more than one person who carried out the action.
Firstly, and most commonly, all 'être verbs' need to agree. Let's look at the verb tomber (to fall). It's a verb of movement = être as the auxilary = past participle agrees with the person e.g. I fell - Je suis tombÉ (the person who carried out the action was a male) vs Je suis tombÉE (the person who carried out the action was a female) or nous sommes tombÉS (more than one male carried out the action = we fell; we is plural hence the s) vs nous sommes tombÉES (more than one female carried out the action).
We don't make the verb-person agreement with 'avoir' as an auxilary but we do sometimes make the verb-direct object agreement. Namely, when the direct object of the verb is put in front of the verb. Let's look at these examples: j'ai achetÉ les fleurs (I bought the flowers) - the direct object (les fleurs) is put after the verb so there is no verb-object agreement vs les fleurs, je les ai achetÉS (the flowers, I bought them) - the direct object (still the flowers) is put before the verb so we agree it with the verb ie we have the verb-object agreement. Notice that we do not agree the person carrying out the action but the direct object.
Here's and equation to help you remember the formation of the Perfect tense in French:
Choose the auxilary (Avoir or Etre verb?) + put the auxilary in the present tense and in the person needed + add the past participle + agree, if necessary
Avoir/ Etre + past participle + agreement, if necessary