As you may already know, there are 10 tenses/moods in Sanskrit (11 if you include the tense that is used only in the veda). Of these, you will see a similarity in the forms of 4 tenses/moods since they are all derived in a specific manner. Likewise forms of the remaining 6 tenses/moods are all derivable similarly.
If you take up verbs of the first group (called bhvaadi-gana भ्वादिगणः) and examine their conjugations in the 4 tenses/moods
लट् (present tense),
लोट् (imperative mood),
लङ् (past tense) and
विधिलिङ् (imperative/potential mood) (excuse me I don't have the best translations for these tenses/moods), they all follow a specific pattern. These are called saarva-dhatuka-lakaraaha सार्वधातुकलकाराः Knowing the 9 forms (singular, dual and plural for each of the first, second and third persons) will enable you to guess forms for other verbal roots in these 4 tenses/moods. As long as you are aware of certain rules that apply modifications to the verbal base, you should be able conjugate all forms with ease. The common factor for conjugating verbs in these 4 tenses/moods is that there's a suffix that interposes itself between the verbal root and the tense. For e.g
भू + लट् - भू is the verbal root and लट् is the lakara for present tense
भू + तिप् - लट् is replaced with तिप् for first person singular
भू + शप् + तिप् - शप् is this new suffix
भू + अ + ति - The indicatory letters are dropped
भो + अ + ति - A modification to the verbal base
भवति - When two vowels are next to each other, they are modified accordingly.
If you pick verbs from a different group (say the second one called adaadi-gana अदादिगणः), their conjugations will differ only in that the intervening suffix between the verbal base and the person/number suffix changes.
Of course, there are always exceptions to these generic statements and only a study of Panini's grammar will equip one with the tools to conjugate any verbal root for any tense/mood.
The second category is the remaining 6 tenses/moods are called aardha-dhatuka-lakaraaha आर्धधातुकलकाराः They are
लिट् (some kind of past tense)
लुट् (some kind of future tense)
लृट् (some kind of future tense)
लुङ् (some kind of past tense, probably aorist if my memory serves me right)
लृङ्(impossibility of action)
In conjugating verbs for these tenses/moods, it should be noted that a very specific suffix falls in place between the verbal root and the suffix that indicates person/number. For e.g
भू + लुट् - भू is the verbal root and लुट् is the lakara for future tense.
भू + तिप् - लुट् is replaced with तिप् for first person singular
भू + तास् + तिप् - A new suffix तास् appears between भू and तिप् This suffix is the same, regardless of which verbal root you pick (from any of the 10 groups/गणाः)
भू + तास् + डा - तिप् is replaced with डा
भू + इतास् + डा - The verb being सेट्, the augment इट् is added to तास्
भू + इत् + आ - A portion of तास् is elided
भो + इता - A modification to the verbal base
भविता - When two vowels are next to each other, they are modified accordingly.
In these 6 tenses/moods, the knowledge of whether a verb is सेट्/अनिट्/वेट् is very important. In
लिट् the verb is always duplicated and undergoes many modifications. While it is easy to spot the duplication when you come across a conjugated form in reading, deriving the same (or sometimes identifying the verbal root itself) from a verb is non-trivial. In
लुङ् too, there are many rules that govern the modifications to verbal bases and sometimes to suffixes.
Memorizing the conjugations of all verbs in all tenses/moods is a daunting task. One of the main benefits of studying Panini's grammar is that conjugating a verb in any tense becomes trivial with sufficient practice and knowledge of the rules. Someone who hasn't undergone such training may be able conjugate verbs to some extent but will always fall short when special knowledge is required in some places.
A sanskrit verb has just one part. The conjugated part is derived by adding one or more suffixes to the verb.