Vedas do not have any gender based discrimination. When it comes to vaidika dharma and practices both men and women enjoy same freedom. However, after the advent of sanyasa, the society slowly turned towards more male dominated and male chauvinistic in nature. Slowly women were deprived from their basic religious and social rights which were sanctioned by Vedas.
As per Haarita sutra, those women who are Brahma Vaadini have right for initiation, Vedic studies , Fire establishment and bhiksha etc.
द्विऽविधाः स्त्रियो ब्रह्मवादिन्यः सदीवध्वश्च । तत्र ब्रह्मवादिनीनामुपनयनम्योन्धनं वेदाध्ययनं स्वगृहे भिक्शाचर्या चेति ( 21:23)
As per Yama Smriti,
पुरा कल्पे कुमारिणां मोञ्जोबन्धन मिष्यते । अध्यापनन्श्च वेदानां सावित्रीवाचनन्तथा ॥ पिता पितृव्यो भ्राता वा नैनामध्यापयेत् परः |
In olden times tying of girdle was prescribed for girls. So also study of vedas and uttering of Gayatri ( Savitur) preferably from father , brother or uncle. She is allowed for begging alms within her house limits. Yama refers to existence not only in olden days but during his times as well by his sentence पुराकल्पे ( यथैष्यत) तथा (अधुनाऽपि) इष्यते. The word तथा is specially put by Yama by way of comparison between old time and his own time so he says that during his time girls should learn from father, brother etc and she dont need to wear deer skin, bark etc. If Yama had intended to say that girls should not be initiated in his time unlike olden times he would say something like इह कल्पे तासामेवं न कुर्यान्मतिमान् क्कचित् ॥ As he uses verbs in present tense so we can make out that he prescribed rules for his own days and was the case in olden days too.
Gobhila says that Bride should wear the sacred thread when the bridegroom leads her to the altar.
"प्रावृतां यज्ञोपवीतिनीमम्युदानयन् जपेत् सीमोऽदाद् गन्धर्वायेति " ( Gobhila Grhya sutra 2:1:19)
यज्ञोपवीतिनाऽचान्तोदकेन कृत्यम् ( Gobhila Grhya sutra 1:1:2)
Like the women are eligible for vedas, they are eligible for brahmavidya too.
(यथाधिकरः श्रोतेषु योषितां कर्म सुश्रुतःएवमेवानुमन्यम्ब ब्रह्माणि ब्रह्मवादिताम् Yama Smriti )
Brahamchari is one who is engaged in the study of vedas. ब्रह्मचर्येण ब्रह्म वेदः तदध्ययनार्थमाचर्यम् Atharva veda sAyana bhashyam 117.17
A brahmacharin should be married to a brahmacharini.
ब्रह्मचरिणां च ब्रहचारिणीभिः सह विवाह् प्रशस्यो भवति - Pranavavada of maharshi Garga
The Rig-Veda (V, 7, 9) refers to young maidens completing their education as brahmacharinis and then gaining husbands…
Rv. iii (55, 16) mentions unmarried learned and young daughters who should be married to learned bridegrooms.
Without initiated by proper samskara, no one is eligible to utter holy vedic mantras. But women are found to utter them in many of Shrauta and Grhya sacrifices (yajnas).
In the sAkamedha the third of chaturmasa (four monthly) sacrifice the girl uses the Tryambaka's mantra (त्र्यम्बकं यजामहे सुगन्धिं पतिवेदनम्। उर्वारुकमिव बन्धनादितो मुक्षीय माऽमुतः, Shukla Yajurveda Samhita III :60 )
As per Shatapatha Brahmana, The priest and Sacrificer go round the altar thrice from right to left, but maidens go from left to right. While doing so the maidens utter the Tryambaka mantra. ( II : 6 : 2 : 13 )
Yajnikadeva in the paddhati says वचनात् कुमार्या अपि मन्त्रपाठः meaning that maidens should too utter the sacred formulas as prescribed.
In Madana pArijAta the author quotes KAtyayana (KAtyayana samhita p.330) to show that if any samskara before the initiation ceremony is left unobserved, it should be performed with offering of oblation. Since Madana deals with these things under strI samskara section it is clear that women are entitled to initiation.
vAchaknavi is venerated as best among all the learned brahmana present in the court of King Janaka to ask Yajnavalkya, the most learned sage of that time to ask series of questions to ascertain if they should debate with Yajnavalkya or not. Yajnavalkya answered first two questions correctly however on subsequent occasion he did not encourage her to ask any further question. ( Brhd. Up. III : 3 )
References of Vishpalaa being provided with a leg of iron in Khela's battle shows that women were trained in military arts as well. ( RV 1.116.15 , 1.112.10)
As per GovindAnanda, The daughter will have the rights and not her father to perform the funeral and shraddha rites for her mother, provided she is brotherless. ( ShrAddha Kriya Kaumudi p. 464,469)
The mahabhashya shows that women attained fame as Mimamsakas as well. Ladies specialist in Kaashakrtsna were called as Kaashakrtsnaas. It also specially mentioned group of women Grammarians called Aapishalaas. ( 72 )
Sulabhaa was a vidushi (learned) to the extent that she could teach even Janaka. She could not be married as no male of time was equal to her in terms of knowledge therefore she was initiated into brahmavidya. (Mbh. XII : 320 )
In the Varuna PraghAsas ( second four monthly) sacrifice, after havis is placed and when wife leading to the fire for the karambha pAtra oblations. She utters the mantra "प्रघासिनो हवामहे मरुतः ..." [Shukla Yajurveda Samhita III : 45 ] for invoking the Marut's.
In AgnisToma sacrifice ( one of mahayaga) the wife while entering from south door, rubs the rods of the Soma Cart with remaining ghee left after the sAvitrI Homa with the mantra " देवश्रुतौ देवेष्वा घोषतम्प्राची ....." [ Shukla Yajurveda Samhita V: 17 ]
As per Shraddha Kalpataru "स्त्रीणामप्येवम्" etc means " याध्यशेन सम्बन्धेन पित्रुव्यत्वादिना पुरुषाणामेकादशाहादि श्रद्धं, तट्टशेन च सम्बन्धेन स्त्रीणामेतत् कर्तव्यमिति " which shows that women also utter the mantras in the shraddha.
Yajnavalkya says : After offering Ekoddistha and Sapindikarana with mantras. Both should be offered to women also ( Unavimsati Samhita 1 : 2 )
As per Shankha, the daughter too like the son after the samskara, should observe the Ashaucha and perform funeral rites and do pinda dana and celebrate the Ekoddishta for her father. Here too no distinction has been mentioned between son and daughter.
"दुहिता पुत्रवत् कुर्यात् माता-पित्रोस्तु संस्कृता । अशौचमुदकं पिण्डमेकोद्दिष्टम् सदा तयोः"
In the shraddha's the middle cake is to be eaten by the wife ( Patni ) ( Shraddha Manjari )
Devana Bhatta in the Shraddha Kaanda says : If she is desirous of a child , the wife should eat the cake. ( Smritichandrikaa Vol 4 )
From the time of marriage, the householder and wife , children ( son & daughter) or disciple should offer oblations to the Grhya Agni ( Ashvalayana Grhya Sutra 1: 7 : 1 )
Wife should verily offer the oblations to the Grhya fire at morning and evening
"kAmaM gRhyegnau patnI juhuyAt sAyaMprAtarhomau, gRhAH patnI gRhya eShognirbhavatIti" ( Gobhila Grhya sutra 1:3:15 )
Wife is so closely related to Agni that in the case of extinction of the fire, the wife has to do the fasting as a prayashchitta. ( "यदि तूपशाम्येत्....." Ashvalaayana Gruhya sutra 1:7:3 )
One of the commentary of Gobhila sutra states that wife should be given enough education so that she becomes eligible to perform the sacrifice since the sutras do not say that she should offer oblations silently.
न खल्वनधोत्य शन्कोति पतिहोतुम् । न च तुष्णीमित्याह वचनम् ( Gobhila 1:5:15)
Wife desirous of a child should offer the first aahuti in the morning and evening with the prescribed mantra ending with SvAhA.
"......पुनः स्वाहेति पूर्वां गर्भकामा" (Paraskara Grhya sutra 1:9:3)
From above evidences it is clear than women have right to vedas and yajnopavitam which is denied by most of vedantins.