dotnet --info output:
VS Code version:
C# Extension version:
Here is the output of omisharp log.
Starting OmniSharp server at 18/04/2020, 18:00:40
Target: c:UsersluckyDesktopUnity projectsNew Unity ProjectNew Unity Project.sln
OmniSharp server started.
Path: C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0OmniSharp.exe
PID: 12860
Starting OmniSharp on Windows 6.2.9200.0 (x64)
DotNetPath set to dotnet
Located 1 MSBuild instance(s)
1: StandAlone 16.4 - "C:\Users\lucky\.vscode\extensions\ms-dotnettools.csharp-1.21.17\.omnisharp\1.35.0\.msbuild\Current\Bin"
MSBUILD_EXE_PATH environment variable set to 'C:\Users\lucky\.vscode\extensions\ms-dotnettools.csharp-1.21.17\.omnisharp\1.35.0\.msbuild\Current\Bin\MSBuild.exe'
Registered MSBuild instance: StandAlone 16.4 - "C:\Users\lucky\.vscode\extensions\ms-dotnettools.csharp-1.21.17\.omnisharp\1.35.0\.msbuild\Current\Bin"
CscToolExe = csc.exe
CscToolPath = C:\Users\lucky\.vscode\extensions\ms-dotnettools.csharp-1.21.17\.omnisharp\1.35.0\.msbuild\Current\Bin\Roslyn
MSBuildExtensionsPath = C:\Users\lucky\.vscode\extensions\ms-dotnettools.csharp-1.21.17\.omnisharp\1.35.0\.msbuild
MSBuildToolsPath = C:\Users\lucky\.vscode\extensions\ms-dotnettools.csharp-1.21.17\.omnisharp\1.35.0\.msbuild\Current\Bin
Detecting Cake files in 'c:\Users\lucky\Desktop\Unity projects\New Unity Project'.
Could not find any Cake files
Detecting projects in 'c:\Users\lucky\Desktop\Unity projects\New Unity Project\New Unity Project.sln'.
Queue project update for 'c:\Users\lucky\Desktop\Unity projects\New Unity Project\Assembly-CSharp.csproj'
Detecting CSX files in 'c:\Users\lucky\Desktop\Unity projects\New Unity Project'.
Could not find any CSX files
Invoking Workspace Options Provider: OmniSharp.Roslyn.CSharp.Services.CSharpFormattingWorkspaceOptionsProvider, Order: 0
Loading project: c:\Users\lucky\Desktop\Unity projects\New Unity Project\Assembly-CSharp.csproj
Invoking Workspace Options Provider: OmniSharp.Roslyn.CSharp.Services.RenameWorkspaceOptionsProvider, Order: 100
Invoking Workspace Options Provider: OmniSharp.Roslyn.CSharp.Services.ImplementTypeWorkspaceOptionsProvider, Order: 110
Invoking Workspace Options Provider: OmniSharp.Roslyn.CSharp.Services.BlockStructureWorkspaceOptionsProvider, Order: 140
Invoking Workspace Options Provider: OmniSharp.Roslyn.CSharp.Services.EditorConfigWorkspaceOptionsProvider, Order: 200
Configuration finished.
Omnisharp server running using Stdio at location 'c:\Users\lucky\Desktop\Unity projects\New Unity Project' on host 8312.
The referenced project 'Unity.Timeline.Editor.csproj' does not exist.
The referenced project 'Unity.VSCode.Editor.csproj' does not exist.
The referenced project 'Unity.TextMeshPro.Editor.csproj' does not exist.
The referenced project 'UnityEngine.UI.csproj' does not exist.
The referenced project 'Unity.Timeline.csproj' does not exist.
The referenced project 'Unity.CollabProxy.Editor.csproj' does not exist.
The referenced project 'Unity.Rider.Editor.csproj' does not exist.
The referenced project 'Unity.TextMeshPro.csproj' does not exist.
The referenced project 'UnityEditor.UI.csproj' does not exist.
The reference assemblies for .NETFramework,Version=v4.7.1 were not found. To resolve this, install the Developer Pack (SDK/Targeting Pack) for this framework version or retarget your application. You can download .NET Framework Developer Packs at https://aka.ms/msbuild/developerpacks
Failed to load project file 'c:\Users\lucky\Desktop\Unity projects\New Unity Project\Assembly-CSharp.csproj'.
c:UsersluckyDesktopUnity projectsNew Unity ProjectAssembly-CSharp.csproj
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'Unity.Timeline.Editor.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'Unity.VSCode.Editor.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'Unity.TextMeshPro.Editor.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'UnityEngine.UI.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'Unity.Timeline.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'Unity.CollabProxy.Editor.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'Unity.Rider.Editor.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'Unity.TextMeshPro.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1850,5): Error: The referenced project 'UnityEditor.UI.csproj' does not exist.
C:Userslucky.vscodeextensionsms-dotnettools.csharp-1.21.17.omnisharp1.35.0.msbuildCurrentBinMicrosoft.Common.CurrentVersion.targets(1177,5): Error: The reference assemblies for .NETFramework,Version=v4.7.1 were not found. To resolve this, install the Developer Pack (SDK/Targeting Pack) for this framework version or retarget your application. You can download .NET Framework Developer Packs at https://aka.ms/msbuild/developerpacks
Attempted to update project that is not loaded: c:\Users\lucky\Desktop\Unity projects\New Unity Project\Assembly-CSharp.csproj
Miscellaneous file: c:\Users\lucky\Desktop\Unity projects\New Unity Project\Assets\NewBehaviourScript.cs added to workspace
Hi @blackshadow369!
I was with a problem almost equals yours, I solved it changing the TargetFrameworkVersion
Go to the file Assembly-CSharp.csproj
Change "TargetFrameworkVersion" to the latest .NET Framework Dev (Targeting Pack) version (remember to download the latest version with you don't have it haha), in my case 4.7.1 became 4.8, for some reason when Unity created the project it cames with that version.
Hopes it helps you and sorry for any english grammar issues, I still learning english haha.
@SCHUGRWS is correct, through Unity may decide to change .csproj file later, and there can be multiple .csproj files that would need such babysitting (Plugins folder, Editor folder, packages etc.).
Ideally you should install targeting pack for 4.7.1. Best way to do this is: Visual Studio Installer -> <select your latest VS installation> -> Modify -> Individual Components

You can try to install targetting pack manually, but I've had problems with that, from my experience installing them with "big" Visual Studio is the only sure way to make everything work.
Since installing Visual Studio may be too much. You can get the targeting packs and a full MSBuild by installing the "Build Tools for Visual Studio" from https://visualstudio.microsoft.com/downloads/

Going to close this out as by design. The issue is stemming from not having .NET Framework targeting packs installed, which is necessary to target .NET Framework projects.
Most helpful comment
Since installing Visual Studio may be too much. You can get the targeting packs and a full MSBuild by installing the "Build Tools for Visual Studio" from https://visualstudio.microsoft.com/downloads/