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While searching the Net for 'Ashtaadhyaayee chanting' I chanced to scout the following from www.ambaa.com site. Since it contains useful hints to the earnest student of Sanskrit.
www.advaita-vedanta.org

Sanskrit Grammar

(1) Kasika.  edited by Vijayapala Vidyavaridhih, published by Ramlal Kapoor Trust.
is the standard Sanskrit commentary (more than a thousand years old) on the Panini's Ashtaadhyaayee. There are many editions available but the following one is good as it is completely error-free : this edition has only the original Sanskrit text with no translation or commentary. The following two have translations:

(2) Kasika.   edited by Jayasankaralal Tripathi and Sudhakar Malaviya pub by Tara Printing Works, Varanasi. in 10 volumes.
This has two of the best Sanskrit commentaries on Kasika along with a good Hindi translation.

(3) The Ashtadhyayi of Panini   by Srisa Chandra Vasu,  pub by Motilal Banarasidass.
This is a good English translation of Kasika.

(4) Ashtadhyayi of Panini .  edited by Prof. Gopal Dutt Pandey, pub by Chowkhamba Surabharati Prakashan, Varanasi.
This is a small book which has only the Ashtaadhyaayee Sootra - Sutra Patha (with no commentary), but with Anu-Vritti of the Sootra and an alphabetical index of Sootra at the back.

(5) A Dictionary of Sanskrit Grammar .  by Kashinath Vasudev Abhyankar and J. M. Shukla.  pub in Gaekwad's Oriental Series #134.
This is a really superb English explanation of pretty much all Sanskrit grammatical terms. It also gives the numbers of theSootra where the term occurs. It is a must have book.

(6) The Tested Easiest Method of Learning and Teaching Sanskrit . First Book. By Pandit Brahmadattaji Jihnasu. Pub by Ramlal Kapoor Trust.
For those who want to make a gradual entry into the Panian system this is a good book: This book is also available in Hindi. There is a part two also (but only in Hindi.) by Yudhishthira Mimamsaka, same publisher.

(7) Siddhanta Kaumudi .
The Siddhanta Kaumudi is the re-ordering of Panini's Asthaadhyaayee by topic, done by the great grammarian Bhattoji Deekshit about 400 years ago. There are many editions and commentaries of this book, but here are some good ones:

7.1. Siddhanta Kaumudi" with Tattvabodhini commentary edited by Vasudev Lakshman Shastri Panashikar, pub by Chaukhamba Sanskrit
Pratishthan, Delhi. This is a re-print of the famed old Nirnaya Sagar edition It has a good set of appendices (including Dhatau Patha, Sootra Paatha etc) at the back.
7.2. Vyakarana Siddhanta Kaumdi .  edited by Giridhar Sarma Chaturvedi and Parameswarananda Sarma Vidyabhaskar. pub by Motilal Banarasidass. 4 volumes
This edition has two Sanskrit commentaries - particularly the Balamanorama which is the easiest to read and hence the most popular, even though it has some errors.
7.3. The Siddhanta Kaumudi .  by Srisa Chandra Vasu pub by Motilal Banarasidass. For those who prefer English:

(8) Laghu Siddhanta Kaumudi
The most popular Sanskrit grammar text studied today is the Laghu Siddhanta Kaumudi, which as the name implies, is a condensed Siddhanta Kaumudi containing about 1300 of the most important Sootra organized by topic. Here again there are many editions available but the following Hindi translation is absolutely outstanding. This is the first place you can go to when you have a question. It is a must have even if your Hindi is weak.

8.1 Laghu Siddhanta Kaumudi with Bhaimi Vyakhya.  pub by Bhaimi Prakashan, 537, Lajpat Rai Market, Delhi - 110006. 6 volumes.
One should have at least one edition of the Kaasika and one of the Siddhanta Kaumudi.

Dhaatu Paath

Since the Sanskrit language is almost entirely based on the verbal roots, there are many commentaries on the Dhaatu Paath which explain not only the meaning of the Dhaatu but also derive the forms which come from it. Here are some useful books:

1. Krdanta Rupa Mala . pub by The Sanskrit Education Society, Madras. 5 volumes.
This is a superb collection of all the Dhaatu along with it's major forms - not only of the original verbal root but also its causative and desiderative variations. More importantly it gives all the necessary Sootra and references from literature. An extremely useful reference, but unfortunately out of print.

2. Madhaviya Dhatu Vritti .  edited by Vijayapala Vidyavaridhih. pub by Ramlal Kapoor Trust.
This is a very respected Sanskrit commentary on the Dhaatu Paath by the great Vaidik commentator Sayanacharya.

3. Kshira Tarangini .  commentary on the Dhatu Patha by Kshiraswami. edited by Yudhishthira Mimamsaka, published by Ramlal Kapoor Trust.
A very old standard commentary. A well edited work with good footnotes.

Dictionaries, etc.:

The most well-known and original thesaurus of Sanskrit is undoubtedly the Amara Kosa. No collection of Sanskrit books is complete without this one. Of the various editions available I like the following:

1. Namalinganusasana alias Amarakosa .  Amarsimha with Sanskrit commentary, edited by Pandit Sivadatta Dadhimatha and revised by Vasudev Laksmana Panasikara, published by Chaukhamba Sanskrti Pratishthan (reprint of old Nirnaya Sagar edition).
This edition is particularly useful because the commentary gives all the Sootra necessary for the etymology of each word.

2. Amarakosa . edited by Prof AA Ramanathan.  pub by The Adyar Library and Research Centre.  3 volumes.  with South Indian Sanskrit commentaries.
Gives good explanations but without Sootra.

3. Avyaya Kosa: A Dictionary of Indeclinables .  pub by The Sanskrit Education Society, Madras.
This is a special dictionary containing all the indeclinable words in the language in alphabetical order. Very useful and full of good information including Sootra. Volume I of "Laghu Siddhanta Kaumudi" with Bhaimi Vyakhya (mentioned above) also has a very good list of indeclinables.

4. Unadi Kosa . edited by Yudhishthira Mimamsaka. pub by Ramlal Kapoor Trust.
Since not all Sanskrit words can be derived using the Ashtaadhyaayee, the Unadi Sootra are like an appendix to the Ashtaadhyaayee, which explains the etymology of words not covered by Paanini. Some of these are very common words - like Manas. Paanini refers to the Unadi Sootras - see Sootra 3.3.1 - but doesn't go into details. Unadi Sootra are also part of the Siddhanta Kaumudi. Thede are called Unadi because the first affix given is 'un'. This is a well-edited handy book of Unadi Sootra along with good appendices.

5. Gana Patha
In addition to the Dhaatu Patha, which gives the list of verbal roots, this is also another important appendix to Paanini's Ashtaadhyaayee. This gives the lists of nouns referred to by Paanini in various Sootra.

Gana Ratnavali. edited by Pandit Chandradatta Sarma. pub by Ramlal Kapoor Trust.
Here is a good edition with a Sanskrit commentary

6. Apte's Practical Sanskrit-English Dictionary - Revised and Enlarged edition.
The best Sanskrit-English dictionary. There are many different variations of Apte's dictionary but this particular one is the most comprehensive of them all. The original publication is from Japan, but that is expensive ($100). There is an Indian reprint of the Japanese original, by Motilal Banarasidass which is affordable (about $15).

7. A Higher Sanskrit Grammar . by Kale. published by Motilal Banarasidas.
A very good all-encompassing reference book in English It also contains some Paanini Sootra as footnotes.

8. Student's Guide to Sanskrit Composition .  by V. S. Apte.  re-published by Chowkhamba Sanskrit Series Office, Varanasi.
Another great work by Apte.

9 Sabda Manjari and Dhatu Rupa Manjari . pub by R. S. Vadhyar & Sons, Palghat.
For a quick guide (without Sootra) to nominal declensions and verbal conjugations. These have a few errors but still handy references.

Geetaa, etc.

Like all Geeta Press books, these are of a high quality and also inexpensive. Read the Raamaayan first. It is much easier.

(1) The best Bhagawad Geeta book for Sanskrit commentaries is Srimad Bhagawad Gita,  edited by Wasudev Laxman Sastri Pansikar, pub by Munshiram Manoharlal (re-print of the old famed Nirnaya Sagar edition). It is well edited with seven commentaries and is really worth having.

(2) For chanting purposes, a very good large type error-free Geetaa book is the Sri Pancha Ratna Geeta by Geeta Press (book #21). This has Vishnu Sahastra Naam Stotram also.

(3) For those who want to practice reading Sanskrit with the help of a good English (or Hindi) translation, it is  recommended that the following Gita Press books :
--Srimad Valmiki Ramayana" in 2 volumes with English translation - Book numbers 452, 453. The Hindi version is book numbers 75, 76.
--Srimad Bhagavata Mahapurana" in 2 volumes with English translation - Gita Press book numbers 564, 565. The Hindi version is book numbers 26, 27.

(4) Bhatti Kavyam.
Bhatti was a very great grammarian who wrote his version of the Raamaayan with the intention of teaching Paanini Sootra. No course on Paanini's Ashtaadhyaayee can be complete without Bhatti. It's not for beginners. A must read for serious students of Paanini. Many editions are available.

Sanskrit Grammar by Ramakrishna Gopal Bhandarkar, in 2 parts.
Sanskrit Grammar by MatakariShastri in 2 Parts, pub by Swadyaya Prakashan, Pardi, Gujarat. (Pt. Satavalekar)

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