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(Google tells me the plural of "lemma" could be "lemmas" or "lemmata", so i chose the latter. Please tell me if it's wrong.)
From reading online and offline sources, I've gathered that lemmata are mostly semantic while roots are mostly morphological. Following this, languages should have some way of taking roots and assigning lemmata to them, giving each root a meaning.
However, I can't find any consistent sources for how a language would assign meaning this way. All I can find are word lists, which would be great if I wasn't trying to design roots with lemmata attached. Is there a process for assigning meaning to roots I should follow?

Sidenote: The language I'm working on is meant to be a PIE-analogue, and I've been looking at PIE roots on Wikipedia. The lists weren't well suited to my goals but did tell me some things about how derivation affects meaning.

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There are a few different methods, and an actual natural language may use a combination of them.

The simplest approach would be to use roots directly, and to require analytic constructions to derive ways of discussing additional senses. This is arguably what something like Toki Pona aims for, but is not very naturalistic, as people will naturally settle on conventional constructions for certain things and it will gradually involve into the next system.

Then you can have synthetic systems that use derivational affixes to derive senses from roots. This will require various derivational affixes that can be attached to roots to form additional lemmata. This will typically include affixes for changing parts of speech (e.g. forming agent nouns from verbs, forming fientive verbs from adjectives, or adjectives of similarity from nouns, amongst many, many others), as well as various affixes for deriving lemmata within the same part of speech (e.g. forming collective nouns, causative verbs, or superlative adjectives).

Multiple such derivational affixes can be applied, although different languages will tolerate chains of differing lengths. Polysynthetic languages allow extremely long chains of affixes, some of which may be derivational, others inflectional.

Related to this is composition which involves forming compounds. These consist of two (or more) parts, these are typically stems, and so may consist of a root (and possibly derivational affixes), with the entire compound inflected as a single unit. Some languages with extensive composition do form compounds of fully inflected lemmata (look up Sanskrit Compounds if you want to see an especially extensive system). This is a common origin for derivational affixes used in synthetic formations.

Then you have apophonic, templatic, or internal formations. These derive lemmata from roots (or each other) through changes internal to the root (or stem) itself, especially changes to the vowels (apophony strictly refers only to changes to vowels).

A few case studies then:

  • Finnish is an agglutinative language and does almost all of its derivation using synthetic or compositional means. Roots are generally bisyllabic and may receive a number of derivational affixes to form a stem (which may then be compounded with one or more other stems) before receiving inflectional endings.
  • Ancient Greek is a fusional language and active derivational processes are generally similar to those in Finnish (i.e. new formations are almost exclusively synthetic or compositional). It does however preserve a wide array of related lemmata derived from roots with varying vowels because at an earlier stage (PIE and shortly afterwards) it made significant use of apophony. This is why a verb like phérō "I bear" with a root "pher" has some related nouns beginning "phor" as in phorós "tribute", and most prominently in the suffix *-phóros "-bearing".
  • Biblical Hebrew, like most Semitic languages, uses templatic derivation as its primary method. In Biblical Hebrew a root consists solely of consonants, with no vowels, and so cannot be used on its own. To form a lemma from a root, the root must be inserted into a pattern or template telling you what vowels to insert where (sometimes alongside affixes). The root q-ṭ-l has senses related to killing and is commonly used to illustrate different patterns. The most common verbal patterns (and the ones that survive into Modern Hebrew) are the active qaṭal, intensive qiṭṭel, causative hiqṭil, reflexive hitqaṭṭel, (medio)-passive niqṭal, passive quṭṭal, and passive causative huqṭal. There are also various noun & adjective patterns, some of which interrelate to each other or to verbal patterns and which often have significant semantic overlap.

As you're interested in PIE, it's worth noting that it uses a somewhat mixed system. Traditionally it's viewed as synthetic and compositional, with a note that apophony does occur, with the applied affixes affecting the vowel "grade" of the stem to which they're applied, but there is a minority arguing that it is better described as a templatic system.

So, let's illustrate how this works by looking at a single root, conventionally cited in the e-grade: **deyḱ- "to indicate", but from a templatic perspective, arguably better represented as **d_yḱ- for which the following forms can be solidly reconstructed (alongside several other less certain formations):

  • *déyḱti: an imperfective (present) verb "to be indicating" formed directly from the accented e-grade of the root with athematic imperfective inflectional endings (and with further apophony between distinct inflectional forms).
  • *dḗyḱst: a perfective (aorist) verb "to point out" formed from the accented lengthened e-grade of the root with a derivational *-s- suffix with athematic perfective inflectional endings (and with further apophony between distinct inflectional forms).
  • *doyḱéyeti: a causative imperfective verb "to demonstrate" formed from the unaccented o-grade of the root, with a derivational -éy- suffix and thematic imperfective inflectional endings.
  • *déyḱs: an animate noun "indicator" formed directly from the accented e-grade of the root with athematic animate inflectional endings (and with further apophony between distinct inflectional forms).
  • *díḱeh₂: a feminine noun "indication" formed directly from the accented zero-grade of the root with feminine inflectional endings.
  • *diḱtós: an adjective "shown" formed from the unaccented zero-grade of the root with a derivational *-t- suffix and thematic inflectional endings.

This illustrates the extensive use of a apophony within PIE, as well as a small number of derivational suffixes, some of the most widely attested in PIE (the *-s- for forming athematic perfective verbs from imperfective roots or stems, the *-éy- for forming causative imperfective verbs from verbs, and the *-t- for forming thematic adjectives with a passive sense).

You are also correct on the plural(s) of lemma. The form lemmata is accurate to the Greek, whilst lemmas is the regular English plural.

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