This article will show a way to fetch the values of a picklist field using few approaches. 

1. UI-API – Using the run time api to fetch the picklist values without writing APEX.
   • getPicklistValues – use this wire adapter to get the picklist values for a specified field. (Will require using @salesforce/schema module to describe the fieldName)
   • getPicklistValuesByRecordType – allow more dynamic context.
2. APEX – Using Global describe methods to access object metadata.

Final Result output

Lets get started

HTML – display the values

<template>
    <div class="slds-var-p-around_medium">
      <label for="picklist">{picklistField.label}</label>
      <select id="picklist" class="slds-select" onchange={onChangeFieldValue}>
            <template for:each={picklistField.items} for:item="option">
                <option key={option.name} value={option.name} selected={option.default}>{option.label}</option>
            </template>
     </select>
    </div>
</template>

UI-API – Option 1

using getPicklistValues with dynamic context object and a defined schema field.

import { LightningElement, api, wire } from 'lwc';
import { getObjectInfo } from 'lightning/uiObjectInfoApi';
import { getPicklistValues } from 'lightning/uiObjectInfoApi';
import INDUSTRY_FIELD from '@salesforce/schema/Account.Industry';

export default class Example extends LightningElement {
    @api objectApiName; // Account
     _picklistField;
    // get context object metdata
    @wire(getObjectInfo, { objectApiName:'$objectApiName' })
    objectInfo;

    @wire(getPicklistValues, { 
           recordTypeId: '$objectInfo.data.defaultRecordTypeId', 
           fieldApiName: INDUSTRY_FIELD 
    })
    IndustryPicklistValues;

   // get the picklist field details
   get picklistField(){
        if(this.objectInfo.data && this.picklistFields.data){
           const picklistOptions = this.IndustryPicklistValues.map(item => ({
                        name: item.value,
                        label: item.label
           }));
           this._picklistField = {
               name: this.objectInfo.data.fields.Industry.apiName,
               label: this.objectInfo.data.fields.Industry.label,
               items: picklistOptions
            }
        }
        return this._picklistField ? this._picklistField : false;
}

UI-API – OPTION 2

Using getPicklistValuesByRecordType with dynamic context and dynamic fieldName assignment

import { LightningElement, api, track, wire } from 'lwc';
import { getObjectInfo } from 'lightning/uiObjectInfoApi';
import { getPicklistValuesByRecordType } from 'lightning/uiObjectInfoApi';

export default class Example extends LightningElement {
    @api objectApiName; // Account

    @wire(getObjectInfo, { objectApiName:'$objectApiName' })
    objectInfo;
    @track picklistField;

    @wire(getPicklistValuesByRecordType, { 
        objectApiName: '$objectApiName', 
        recordTypeId: '$objectInfo.data.defaultRecordTypeId'
    })
    picklistFields;
    
    connectedCallback(){
        // sets field name
        const selectedFieldName = 'Industry';
        this.getPicklistOptionsByFieldName(selectedFieldName);
    }
   // assign a field name dynamically and filter the picklist fields list
   getPicklistOptionsByFieldName(fieldName){
        if(this.objectInfo.data && this.picklistFields.data){
            const picklistOptionsList = this.picklistFields.data.picklistFieldValues[fieldName].values;
            const picklistOptions = picklistOptionsList.map(item => ({
                        name: item.value,
                        label: item.label
                    }));
            this.picklistField = {
               name: this.objectInfo.data.fields[fieldName].apiName,
               label: this.objectInfo.data.fields[fieldName].label,
               items: picklistOptions
            }
        }
   }
}

APEX

The Old school way, we are using Apex class and schema describe methods to fetch the values.

public with sharing class PicklistOptionsController {
/* 
 * @description: will return the field details with the option values. 
 * @params: objectApiName and picklist fieldName 
*/
   @AuraEnabled
   public static string getPicklistOptionsByFieldName(String objectApiName, String fieldName){
    Map<String,Object> fieldDetails = new Map<String,Object>();   

         // describe field
         Schema.DescribeSObjectResult contextObject = Schema.getGlobalDescribe().get(objectApiName).getDescribe();
         Schema.DescribeFieldResult fieldDescribe = contextObject.fields.getMap().get(fieldName).getDescribe();

         // Check if it's actual a picklist Field type
         if(fieldDescribe.getType() == Schema.DisplayType.PICKLIST) {
            // gets the picklist values from the field
            List<Schema.PicklistEntry> picklistRawOptions = fieldDescribe.getPicklistValues();
            List<Map<String,Object>> picklistOptions = new List<Map<String,Object>>(); 
            if(picklistRawOptions.size()>0){
                for(Schema.PicklistEntry entry: picklistRawOptions) {
                    // get all options from the picklist option
                    Map<String,Object> optionDetails = new Map<String,Object>{
                        'label' => entry.getLabel(),
                        'name' => entry.getValue(),
                        'active' => entry.isActive(),
                        'default' => entry.isDefaultValue()
                    }; 
                    picklistOptions.add(optionDetails);
                }
            }
            // put field picklist values with the field label
           fieldDetails = new Map<String,Object>{
                'label' => fieldDescribe.getLabel(),
                'name' => fieldDescribe.getName(),
                'items' => picklistOptions
            };  
        }
        else {
            System.debug('This field is not a picklist field - ' + fieldDescribe.getName());
        }
    return JSON.serialize(fieldDetails);
   }
}

Reference your apex in your Javascript

Calling the apex method to retrieve the picklist values and to assign them into the UI Select input.

import { LightningElement,api,track } from 'lwc';
import getPicklistOptionsByFieldName from '@salesforce/apex/PicklistOptionsController.getPicklistOptionsByFieldName';

export default class Example extends LightningElement {
    @api objectApiName; // Account
    @api recordId;
    
    @track picklistField;
    @track selectedFieldName; 

    connectedCallback(){
        this.selectedFieldName = 'Industry';
        this.fetchPicklistValues();
    }

    // fetch values of picklist field options
    fetchPicklistValues(){
        getPicklistOptionsByFieldName({ 
            objectApiName : this.objectApiName,
            fieldName : this.selectedFieldName
        }).then( result => { 
            this.picklistField = JSON.parse(result);
        }).catch(error =>{
            console.log('error ' + error.body.message);
        });
    }

    onChangeFieldValue(event){
        const selectedFieldValue = event.target.value;
        console.log('selectedFieldValue ' + selectedFieldValue);
    }
}

Resources:

• PicklistEntry Class – Documentation